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The physics of manganites: Structure and transport
Myron B. Salamon
Department of Physics and Materials Research Laboratory, University of Illinois
at Urbana-Champaign, Urbana, Illinois 61801
Marcelo Jaime
MST-NHMFL, MS E536, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
(Published 29 August 2001)
The fundamental physical properties of doped LaMnO3, generically termed ‘‘manganites,’’ and much
of the underlying physics, were known more than 40 years ago. This article first reviews progress made
at that time, the concept of double exchange in particular, and points out the missing elements that
have led to a massive resurgence of interest in these and related materials. More recent research is
then described, treating first the ground states that emerge as divalent atoms are substituted for
trivalent La. A wide range of ground states appear, including ferromagnetic metals, orbital- and
charge-ordered antiferromagnets, and more complex stripe and spin-glass states. Because of the
interest in so-called colossal magnetoresistance that occurs in the ferromagnetic/metallic composition
range, a section is devoted to reviewing the atypical properties of that phase. Next the
high-temperature phase is examined, in particular, evidence for the formation of self-trapped small
polarons and the importance of Jahn-Teller coupling in this process. The transitions between the
high-temperature polaronic phase and the ferromagnetic and charge-ordered states are treated in a
fourth section. In each section, the authors stress the competition among charge, spin, and lattice
coupling and review the current state of theoretical understanding. They conclude with some
comments on the impact that research on these materials has on our understanding of doped oxides
and other strongly correlated electronic materials.
VII. Implications
I. Introduction
II. Background
A. Early experiments
B. Early theoretical ideas
1. Double exchange
2. Magnetic and transport properties
III. Low-Temperature States
A. Parent compounds
B. Doped compounds
1. Charge ordering at half filling: n⫽⬁
2. Charge ordering at half filling: n⫽2
3. Other fractional doping levels
4. Theoretical situation
C. Ferromagnetic regime, 3D materials
1. Magnetic properties
2. Electrical transport properties
3. Thermal properties
a. Thermal conductivity of 3D materials
b. Thermal conductivity of layered materials
c. Thermoelectric properties and heat
IV. High-Temperature Behavior
A. Polaron effects—3D materials
B. Polaron effects—layered manganites
V. Ferromagnetic/Paramagnetic Phase Transitions
A. Theoretical background
B. Nature of the phase transition
C. Two-phase behavior
D. Phase-separation models
E. Hall effect in the transition region
VI. Charge and Orbital Ordering Transitions
A. 3D manganites
B. Layered manganites
Colossal magnetoresistance manganites (i.e., based on
LaMnO3 and relatives) are known for the unusually
large effect an external magnetic field has on their ability to transport electricity and heat. While many of these
effects were known more than 50 years ago (van Santen
and Jonker, 1950), an appreciation of the size of these
effects is a more recent development (Jin, Tiefel et al.,
1994). A sensitivity of this magnitude is not observed in
any bulk metallic system where, mostly at low temperatures, magnetoresistance arises from a field dependent
electronic mean free path. This effect is measurable only
in those metals that have a large mean free path in zero
field, i.e., very clean systems, where the magnetic field
reduces the mean free path by inducing the electrons to
move in orbits. Even then, the resistivity change is usually limited to a few percent in practical magnetic fields.
Giant-magnetoresistance multilayer metallic films show
a relatively large sensitivity to magnetic fields. The
mechanism in these films is largely due to what is known
as the spin-valve effect between spin polarized metals. If
an electron in a regular metal is forced to move across a
spin-polarized metallic layer (or between spin-polarized
layers) it will suffer spin-dependent scattering. If the
electron was initially polarized parallel to that of the
layer the scattering rate is relatively low; if originally
polarized antiparallel to that of the layer the scattering is
high. (The reverse is also possible.) The effect of an ex-
©2001 The American Physical Society
M. B. Salamon and M. Jaime: Manganites: Structure and transport
ternal field is to increase the ratio of the former events,
reducing the latter, by aligning the polarization of the
magnetic layer along the direction of the external field.
This effect is a few tens of percent, and has the very
important advantage of not being limited to low temperatures. Spin-valve devices have been used in the
magnetic storage industry for several years now, in the
form of magnetoresistive read heads. While the physical
mechanism that produces the magnetoresistance is well
understood, the technological challenges involved in the
production of small devices of high sensitivity are the
bottleneck of an industry ever hungry for smaller-fasterbetter sensors.
Magnetic systems of great potential are those with a
limited ability to transport electricity in zero field, resulting from competing dissimilar ground states. In these
systems, magnetic fields produce truly dramatic effects
by inducing phase transitions or increasing the temperature of already existing phase transitions. Examples of
this behavior are found in EuO, pyrochlores, and manganites. Also small gap semiconductors like Ce3Bi4Pt3
and YbB12 can be considered in this category, although a
detailed discussion of their ground state is beyond the
aim of this review. The limitation of semiconductors in
general is that room temperature is a large energy scale.
If the gap is large enough to be in the intrinsic conduction limit at room temperature, the magnetic fields required to see changes are unattainable. If the gap is
small and magnetic fields within reach, then the effects
are limited to very low temperatures (Jaime et al., 2000).
In a sense manganites are ideal compounds for magnetic
sensor devices, since the two competing ground states
are metallic and semiconducting, respectively, and because the energy scale of the phenomenon produces the
most interesting effects, i.e., the metal-insulator transition and hence the maximum sensitivity to external
fields, at temperatures close to room temperature (Jin,
McCormack et al., 1994). See Fig. 1 for recent data on
single crystals. It is exactly these features that inspired a
tremendous effort from experimentalists and theoreticians to reexamine these materials and to understand
the mechanisms involved. Despite this effort neither
practical applications nor a satisfactory understanding of
the physics of manganites has yet emerged. In the meantime, our perception of the manganites has changed
radically. They have been redefined from what seemed a
straightforward application in the magnetic-storage industry to a challenge of colossal dimensions from the
condensed-matter physics point of view.
Soon after the rediscovery of these materials, theoreticians pointed out that the theoretical framework used
in the past to explain their behavior qualitatively does
not survive when confronted with a quantitative analysis
(Millis et al., 1995). With an understanding of the complexity of the problem came the realization of the
uniqueness of manganites as a test field for condensedmatter physics theories. Manganites are prototypical of
correlated electron systems where spin, charge, and orbital degrees of freedom are at play simultaneously, and
where classical simplifications that neglect some interacRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 1. The temperature dependence of ␳ xx (H,T) of a
La2/3(Pb, Ca)1/3MnO3 single crystal at various fields H. Inset:
M H (T) in the transition region. From Chun, Salamon,
Lyanda-Geller et al., 2000.
tions to study others in detail simply do not work. We
are convinced that the comprehension of complex problems posed by the manganites has the potential to produce a substantial advance in the theory of condensed
matter, and will eventually shed light into other unsolved problems, for example, high-T c superconductivity. Several excellent review articles (Ramirez, 1997;
Coey et al., 1999; Tokura and Tomioka, 1999; Rao et al.,
2000) and edited books (Rao and Raveau, 1998; Tokura,
2000) are available in the literature now which discuss
the properties of manganites, mostly from a materials
science point of view. A review of the optical properties
of manganites and related materials has recently appeared (Cooper, 2001). Our focus is different since we
choose to emphasize the physics and mechanisms that
determine the properties of manganites in different temperature regions. This approach originated in the realization that the most relevant energy scale in these complex systems is the temperature at which the magnetic
ordering is observed, and that the interplay of magnetic,
charge, and spin degrees of freedom changes qualitatively and quantitatively depending on whether the
temperature is much smaller than, similar to, or much
larger than the ordering temperature, irrespective of the
In this review we focus on the present understanding
of what we think are the most remarkable physical properties of manganites, observed by state of the art experiments and interpreted with modern theories. We have
organized the discussion into three main modules: (i)
low temperatures, or energy scales much smaller than
that responsible for the metal-insulator, magneticparamagnetic phase transition, (ii) high temperatures, or
M. B. Salamon and M. Jaime: Manganites: Structure and transport
energy scales much bigger than the energy scale of magnetic interactions, and (iii) intermediate temperatures,
or energy scales comparable to the interactions where
semiconducting and metallic ground states are both
highly unstable and susceptible to such external parameters as temperature, magnetic field, electrical field,
crystalline disorder, dimensionality, and doping. Before
the discussion of the physical properties we introduce
the subject with a brief historical summary of both experimental results and theory developments since the
discovery of manganites in the early 1950s. We end the
review with a discussion of the implications of manganites.
A. Early experiments
Manganese compounds of composition AMnO3 (A
⫽La, Ca, Ba, Sr, Pb, Nd, Pr) crystallize in the cubic structure of the perovskite mineral CaTiO3 (Verwey et al.,
1950), and will hereafter be referred to as manganites.
Depending on the composition they show a variety of
magnetic and electric phenomena, including ferromagnetic, antiferromagnetic, charge, and orbital ordering. If
the site A is partially occupied by two different atoms,
one trivalent as, for example, La and one divalent as Ca,
then Mn3⫹ and Mn4⫹ coexist in the samples and the
compounds show different behavior as the temperature
is changed. The different phase transitions that the
mixed manganites display (metal-insulator; charge, orbital, and spin degree of freedom; order-disorder transitions) are sensitive to external parameters such as the
pressure and magnetic field. The large sensitivity of the
transport properties to external magnetic fields has been
optimized recently, and because of this manganites are
also known as colossal magnetoresistance (CMR) oxides
(Jin et al., 1994). Some manganese compounds crystallize in quasi-two-dimensional (2D) structures (Ganguly,
1984), called layered manganites. Their physical properties, also related to the presence of both Mn3⫹ and Mn4⫹
ions placed in the center of an oxygen octahedron in the
samples as in the case of the 3D compounds, are in addition anisotropic because the MnO2 planes, where the
magnetic correlations and electrical conductivity take
place, are isolated by two AO planes (Moritomo et al.,
1996). Due to these properties manganites and layered
manganites have attracted a great deal of attention during the last years. Manganites belong to the group of
highly correlated systems where charge, spin, and lattice
degrees of freedom are intimately interrelated and have
the potential to help us improve our understanding of
complex systems.
The study of the properties of manganites started 50
years ago, soon after Jonker and van Santen discovered
a striking correlation between the Curie temperature
(T C ), saturation magnetization (M S ), and electric resistivity (␳) in samples of La1⫺x A x⬘ MnO3, where A ⬘
⫽Ca2⫹, Sr2⫹, and Ba2⫹, when measured as a function of
x (Jonker and van Santen, 1950; van Santen and Jonker,
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
1950). Their polycrystalline samples of composition
La0.7Sr0.3MnO3, Fig. 2, showed maximum values T C
⬇370 K, M S ⬇90 G/g, corresponding to full polarization of all 3d electrons present in the sample, and electrical conductivity ␴ max⬇300 ⍀ ⫺1 cm⫺1, comparable to
single-crystal samples grown more recently (Tokura
et al., 1994). A significant research effort followed soon
after which included the study of the cobaltites (Jonker
and van Santen, 1953) and low-temperature measurements in manganites such as the specific heat, magnetization, dc and ac resistivity, magnetoresistance, magnetostriction, I-V curves, dielectric constant, Seebeck
effect, and Hall effect (Volger, 1954; Jonker, 1954). The
conclusions arrived at by these early researchers were
quite close to those reached by careful examination of
results in recent experiments, a surprising fact considering the time elapsed, the progress in sample preparation
techniques (direct consequence of more than a decade
of high-temperature superconductivity research), and
the advances in our understanding of condensed-matter
Some of the most relevant conclusions reached by researchers in the 1950s are the following:
After studying the correlations between crystal
structure and the Curie temperature and finding
that different samples with the same lattice constant had different Curie temperatures they concluded that a picture of simple exchange interaction could not explain the ferromagnetic transition
temperature in manganites (Jonker and van
Santen, 1950). We now know that the relevant parameter in determining T C is not the distance between manganese, but the angle of the Mn-O-Mn
bond. This is often characterized by the tolerance
factor f⫽( 具 r A 典 ⫹r O)/ 关 &(r Mn⫹r O) 兴 , which compares the Mn-O separation with the separation of
oxygen atom and A-site occupant. These distances
are approximated by the ionic radii of the constituent atoms, suitably averaged and f⫽1 for spherical
atoms packed in the perovskite structure.
(b) They found that samples of composition x⬇0.3
showed maximum Curie temperature and minimum electrical resistivity, and uncovered a linear
relationship between the magnetoresistance and
the magnetization of the specimens, concluding
that magnetism and electrical conductivity were
definitively correlated, Fig. 3 (Volger, 1954).
(c) They established that both divalent element content and oxygen stoichiometry determined the
Mn4⫹ content in the samples.
(d) Their alternating current measurements showed
frequency dependence and a magnetoresistance
that decreased with the applied voltage. These results clearly pointed to an inhomogeneous phase,
and they proposed a model of metallic grains surrounded by a high-resistance intergrain material.
We believe today that inhomogeneity is intrinsic to
the manganites and may play a very important role
in their physics.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
La1⫺x Srx MnO3 vs composition: (a) Curie
temperature ⌰ and (b) saturation magnetization I S at 90 K vs Sr content in percent. (c)
Resistivity ␳ vs Sr content x. Crosses in (a)
and (b) indicate nominal compositions. From
Jonker and van Santen, 1950, and van Santen
and Jonker, 1950.
From Seebeck effect measurements they concluded
that the ferromagnetic ordering had a rigorous influence upon the band structure.
The Hall effect proved extremely difficult to observe, implying extremely low mobility (mean free
path of the order of interatomic distances), and was
anomalous in sign. They attributed these to mixed
conductivity, but we know today that they are also
the signature of charge localization and small polaron conduction.
FIG. 3. Early data on the dc magnetoresistance ⌬␳/␳ of
La0.7Sr0.3MnO3 as a function of the magnetization at (a) room
temperature and (b) 77 K. From Volger, 1954.
Not long after those pioneering experiments, Wollan
and Koehler (1955) published an extensive neutron diffraction study of the series La1⫺x Cax MnO3, identified
the type of magnetic order displayed by the end compounds (Fig. 4), and built the first magnetic structurebased phase diagram for the manganites. This early
phase diagram closely matches our modern versions
(Schiffer et al., 1995). Almost simultaneously (Jonker,
1956) we find the first attempts in the literature to correlate the crystalline structure with the magnetic properties from the point of view of a then-new magnetic interaction proposed by Zener (1951), the so-called
double exchange interaction. According to this picture,
in the configuration Mn3⫹ –O2⫺ –Mn4⫹, the easy simultaneous transfer of an electron from Mn3⫹ to O2⫺ to Mn4⫹
causes at the same time a high electrical conductivity
and, by the tendency of the traveling electron to retain
its spin orientation, a parallel orientation of the mag-
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 4. Difference between neutron-scattering intensity at 4.2
K and room temperature: (a) LaMnO3 showing type-A ordering (inset); (b) CaMnO3, indicating type-G order. From Wollan and Koehler, 1955.
netic moments of the Mn3⫹ and Mn4⫹ ions. Jonker also
found that as the structure of the samples approaches a
stable cubic perovskite defined by a tolerance factor f
close to unity (Jonker and van Santen, 1950), the saturation magnetization of the specimen approaches the
spin only contribution of all Mn ions. Jonker came just
short of concluding what these data cry for, i.e., closer to
cubic structure⇒closer to colinear Mn3⫹ –O2⫺ –Mn4⫹
bonds⇒stronger double exchange interaction.
Further progress came somewhat later when the
group at Manitoba accomplished the first growth of high
quality millimeter long single-crystal manganites of composition (La, Pb)MnO3 (Morrish et al., 1969). In a series
of five papers they confirmed the results previously obtained in ceramics and improved the understanding of
the physics (Leung et al., 1969; Searle and Wang, 1969).
By that time the non-Heisenberg nature of the ferromagnetic transition was established and a phenomenological model based on a strongly spin-polarized conduction band was proposed to explain it, with the results
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 5. Comparison of magnetization and resistivity models
with experiment: (a) Reduced magnetization M(T)/M(0) as a
function of the reduced temperature T/T C . The experimental
points are taken from Leung et al., 1969. The dashed curve is
calculated in the molecular-field approximation, while the solid
curve is calculated with the Searle and Wang model. (b) Similar comparison for the normalized resistivity vs T/T C . The
solid curve is calculated using the experimental magnetization.
From Searle and Wang, 1969.
shown in Fig. 5(b) (Searle and Wang, 1969). The one
thing that is missing in this whole period is the addition
of the electron-phonon interaction. Even though the lattice was clearly linked to the occurrence of ferromagnetism and the metallic phase (Jonker, 1956), a direct
attempt to find changes in the rhombohedral Bragg
angles at the Curie temperature (Oretzki and Gaunt,
1970) seemed to indicate little lattice involvement. Today we know that hints for the lattice involvement are in
fact there, clearly shown by experimental results that
reveal the local structure around Mn ions (Louca et al.,
During the 1970s little progress was made in the understanding of the manganites, as attention was focused
on studies of other interesting magnetic systems like the
Eu chalcogenides (Esaki et al., 1967; von Molnar and
M. B. Salamon and M. Jaime: Manganites: Structure and transport
Kasuya, 1968) and EuO (Penney et al., 1972). The extensive study of these compounds drove the development
of the concept of localization of charge by magnetic polarons, magnetically ordered clusters originated by exchange between bound carriers, and localized spins (Kasuya and Yanase, 1968; Kasuya et al., 1970; Kasuya,
1970a, 1970b). Other key developments include the
work by Mott and Davis (1971) on amorphous conductors and charge localization by crystalline defects. Dormant manganites reappear in the 1980s in a study by a
Japanese team (Tanaka et al., 1982), where oxygen stoichiometry and transport properties were revisited. In
spite of the huge effort and remarkable theoretical advances of the 1970s a convincing explanation for the behavior of manganites remained elusive. Tanaka et al.
could not arrive at a unique interpretation of their results, in part because of what resulted in a second failed
attempt to measure the Hall effect. They proposed for
the first time a plausible interpretation based on charge
localization into a small magnetic or lattice polaron that
conducts by hopping in the paramagnetic region above
T C , but the mechanism producing such localization was
still unknown. They found the resistivity to have a thermally activated form and the Seebeck coefficient to vary
inversely with temperature. The activation energy and
the energy scale associated with the Seebeck coefficient
differ significantly which, as we note below, is a signature of polaronic behavior.
The 3D manganites are the end (n⫽⬁) members of
the Ruddlesden-Popper series (A, A ⬘ ) n⫹1 Mnn O3n⫹1
(Ruddlesden and Popper, 1958). Interest in the properties of small n members was also triggered by the discovery of the behavior of the n⫽⬁ compounds described previously (Gorter, 1963; Bouloux et al., 1981).
Similar to the n⫽⬁ case, intense and systematic research was renewed only after the discovery of large
magnetoresistance (Moritomo et al., 1996; Battle, 1996)
in the n⫽2 member. In these compounds the MnO6 octahedra are arranged in n planes, separated by two
(Ln, A)O layers. Electrical conduction, magnetic correlation, and orbital ordering take place in the MnO2
planes, and their number and separation determine
physical properties.
Even though many of the experimental data have
been available for a long time, one cannot help but notice some lack of perspective in the early interpretations,
characterized by the emphasis on only certain of the
physical mechanisms. In order to improve the understanding of the manganites a much broader point of
view is necessary, one that is able to put together the
whole body of experimental data, including precise crystallographic details, in addition to the double exchange
ideas of Zener (1951), Anderson and Hasegawa (1955),
Goodenough (1955), and de Gennes (1960). The magnetic and lattice polaron ideas of Kasuya (1959), Kasuya
and Yanase (1968), and Mott and Davis (1971), the developments in the theory of small lattice polarons by
Holstein (1959), the narrow-band model of Kubo and
Ohata (1972), and the concept of Jahn-Teller distortion
of the oxygen octahedra surrounding each Mn ion
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
(Reinen, 1979) are essential ingredients. To these we
must add ideas about phase segregation and intrinsic inhomogeneous behavior explored recently by Moreo
et al. (1999). In what follows we shall try to summarize
the present situation and discuss why the understanding
of the physics of manganites is still incomplete and challenging.
B. Early theoretical ideas
1. Double exchange
Soon after Jonker and Van Santen (1950; van Santen
and Jonker, 1950) discovered the strong correlation between ferromagnetism and metallic conductivity in
doped manganites, Zener (1951) offered an explanation
that remains at the core of our understanding of magnetic oxides. Zener noted that, in doped manganese oxides, the two configurations
␺ 1 :Mn3⫹O2⫺Mn4⫹ and ␺ 2 :Mn4⫹O2⫺Mn3⫹
are degenerate and are connected by the so-called
double-exchange matrix element. This matrix element
arises via the transfer of an electron from Mn3⫹ to the
central O2⫺ simultaneous with transfer from O2⫺ to
Mn4⫹. Zener points out that the degeneracy of ␺ 1 and
␺ 2 , a consequence of the two valencies of the Mn ions,
makes this process fundamentally different from conventional superexchange. Because of strong Hund’s coupling, the transfer-matrix element has finite value only
when the core spins of the Mn ions are aligned ferromagnetically, again distinguished from superexchange
which favors antiferromagnetism. As usual, the coupling
of degenerate states lifts the degeneracy, and the system
resonates between ␺ 1 and ␺ 2 if the core spins are parallel, leading to a ferromagnetic, conducting ground state.
Zener estimates the splitting of the degenerate levels to
be given by the ferromagnetic transition temperature
k B T C and, using classical arguments, predicts the electrical conductivity to be
xe 2 T C
ah T
where a is the Mn-Mn distance and x, the Mn4⫹ fraction. This provided a qualitative description of the data
then available.
Anderson and Hasegawa (1955) revisited Zener’s argument, treating the core spin of each Mn ion classically,
but the mobile electron quantum mechanically. Designating the intra-atomic (Hund’s) exchange energy by J
and the transfer matrix element by b, Anderson and
Hasegawa found that Zener’s level splitting is proportional to cos(␪/2), where ␪ is the classical angle between
the core spins. The fundamental result, which has remained a cornerstone of double exchange theory, is that
the effective transfer integral becomes t eff⫽b cos(␪/2).
The energy is lower when the itinerant electron’s spin is
parallel to the total spin of the Mn cores. They also show
that the assumption of classical spins can be avoided if
one replaces cos(␪/2) by (S 0 ⫹1/2)/(2S⫹1), where S 0 is
M. B. Salamon and M. Jaime: Manganites: Structure and transport
the total spin of the two Mn ions and the mobile electron and S is the core spin. Anderson and Hasegawa
made the interesting conjecture that a plot of the inverse
susceptibility versus temperature would exhibit pure
Curie-law behavior at high temperatures (effective T C
⫽0), curving downward as the actual ferromagnetic
transition is approached, lying between that of an ordinary ferromagnet and a ferrimagnet. As we will see, the
observed behavior is somewhat more complicated.
After a gap of some years, de Gennes (1960) revisited
this problem, treating the effect of double exchange in
the presence of an antiferromagnetic background. He
considered a layered material with N magnetic ions per
unit volume, each spin S coupled ferromagnetically to
its z ⬘ neighbors on the same layer with exchange energy
J ⬘ and antiferromagnetically to z neighbors on adjacent
layers with energy J. When the angle between the magnetization of successive layers is ⌰, the exchange energy
can be written as
E ex /N⫽⫺z ⬘ J ⬘ S 2 ⫹zJS 2 cos ⌰.
The double exchange contribution is calculated in the
tight-binding approximation, using the AndersonHasegawa effective transfer integral, to be
E m /N⫽⫺xb ⬘ ␥ k⬘ ⫹b ␥ k cos ⌰/2,
where ␥ k and ␥ k⬘ are, as usual, the sum of phase factors
over the z and z ⬘ nearest neighbors, and x is the Mn4⫹
concentration. The ferromagnetic state is stable at all
temperatures only if the concentration x⭓4JS 2 /b. At
lower concentrations, there is a low-temperature transition to a canted state that extends to x⫽0; the canting
angle between successive planes is given by ⌰ 0
⫽2 cos⫺1(bx/4JS 2 ), reaching ␲ as x→0. When x
⬍2.5JS 2 /b, the canted state gives way to antiferromagnetic alignment along a critical line, while for 2.5JS 2 /b
⭐x⭐4JS 2 /b the intermediate state is ferromagnetic.
The boundary between ferromagnetic and antiferromagnetic intermediate states occurs when the canting angle
is ⌰ 0 ⫽2 cos⫺1(2.5/4)⫽103°. In a concluding section, de
Gennes anticipates current research by considering selftrapping of a carrier by distortion of the spin lattice, an
entity we would now refer to as a magnetic polaron. He
demonstrates that at small values of x, local distortions
of the antiferromagnetic structure always tend to trap
the doped-in charge carrier.
In a remarkably complete treatment, Kubo and Ohata
(1972) considered a fully quantum mechanical version of
a double exchange magnet. They introduced the nowstandard Hamiltonian
i, ␴ , ␴ ⬘
共 Si • ␴ ␴ , ␴ ⬘ 兲 c i†␴ c i ␴ ⬘ ⫹
i,j, ␴
t ij c i†␴ c j ␴ ,
where c i†␴ , and c i ␴ are creation and annihilation operators for an e g electron with spin ␴ on a Mn site and t ij is
the transfer-matrix element. The spin due to t 2g electrons is Si ; ␴ is the Pauli matrix, and J is the intraatomic exchange energy, typically referred to as the
Hund’s-rule exchange energy. Kubo and Ohata argue
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 6. Reduced magnetization M/M 0 vs the reduced temperature T/T C : 䊊, for La0.62Pb0.38MnO3; 䉭, La0.69Pb0.31MnO3;
dashed curve, unmodified, reduced molecular-field curve for
S⫽1.81; solid curve, modified curve for a quartic term coefficient of 0.125. From Oretzki and Gaunt, 1970.
that it is preferable to calculate the motion of a spindown hole moving in a background of spin S⫹1/2 ions.
They transform to a hole version of the Hamiltonian and
use a projection-operator method to consider only the
lowest-energy Hund’s-rule states. Consider two sites
containing a single hole. If both core spins are up, the
hole spin must be down, giving a total spin of 2S⫺1/2.
On the other hand, if the core spins are oppositely
aligned, the configuration belongs to a manifold of total
spin 1/2. The corresponding effective transfer-matrix element ranges between 4t/5 for the aligned case and t/5
in the antiparallel configuration for t ij ⫽t for next neighbors. These may be considered to be the average values
of cos ␪/2 for the quantum version of the problem. The
net result is that the width of the down-spin hole band
broadens in the ferromagnetic limit, while that of the
up-spin hole band narrows, leading to the expected half
metallic ferromagnetic ground state (Irkhin and
Katsnel’son, 1994), in which the Fermi surface lies entirely within one spin subband.
2. Magnetic and transport properties
As a result of the change in bandwidth, the Curie temperature increases as the system magnetizes. Kubo and
Ohata (1972) carried out a mean-field calculation that
takes into account the change in effective Curie temperature, finding that the magnetization increases more
rapidly than it would if it followed the usual Brillouin
function. Figure 5(a) compares the calculated magnetization with that for a conventional mean-field model
with S⫽2; the experimental data shown in the plot are
from Leung et al. (1969) Similar data for other materials
are shown in Fig. 6.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
In addition to the band structure and magnetic properties of the double exchange model, Kubo and Ohata
considered the spin-wave spectrum and the resistivity in
the low-temperature limit. The double exchange mechanism broadens the down-spin hole band, the bottom of
which is at an energy W/2 below the (narrow) up-spin
band, where W is the T⫽0 width of the down-spin band.
If the hole band is filled to a Fermi level ⑀ F , down-spin
states are ␮ ⫽W/2⫺ ⑀ F below any up-spin state. As a
consequence, processes that involve the emission or absorption of a single magnon, and therefore scatter a
down-spin hole into an up-spin state, are suppressed exponentially as exp关⫺(␮/kBT)兴. Kubo and Ohata considered two-magnon processes that do not require a spin
flip, and predicted that the resistivity will vary as T 9/2 at
low temperatures. As we shall see, the T 2 temperature
dependence expected for single-magnon processes is exponentially suppressed at low temperatures in double
exchange magnets.
Polaron transport. More or less concurrent with the
development of double exchange theory, Holstein (1959)
and collaborators studied the properties of charge carriers that are ‘‘clothed’’ in a distortion of the embedding
crystal lattice: the polaron problem. When the hopping
probability is large, the charge carrier hops each time a
neighboring site acquires, via thermal motion, the necessary lattice distortion. In this case, the polaron is said
to be in the adiabatic limit. A characteristic property is
that the electrical conductivity is activated and has the
␴ 共 T 兲 ⬀T ⫺1 exp共 ⫺E ␳ /k B T 兲 ,
where E ␳ is approximately half of the polaron binding
energy. The thermoelectric power, as in a semiconductor,
is inversely proportional to temperature, but the characteristic energy is much smaller than E ␳ . The simple explanation is that the polaron carries with it only its own
chemical potential, but not the energy associated with its
accompanying lattice distortion.
Much discussion has centered on the existence of a
Hall effect in the hopping regime (Entin-Wohlman et al.,
1995). The ‘‘drift velocity’’ in this case is the hopping
distance divided by the time between successive hops
(dwell time). This cannot be construed as a velocity with
which to calculate a Lorentz force. Rather, as Holstein
and collaborators (Friedman and Holstein, 1963; Emin
and Holstein, 1969) demonstrated, the Hall effect arises
through the interference effects induced by the
Aharonov-Bohm phase produced when magnetic flux
threads the loops defined by distinct hopping paths between two sites. The smallest such loop involves one
intermediate site and always gives rise to a negative Hall
coefficient (Holstein, 1973). The resulting Hall coefficient is also activated but, when the elemental hopping
trajectory involves three sites, has an activation energy
E H ⬇2E ␳ /3. These characteristics of polaron transport
were noted above, and will be discussed below, as determinants of polaron-dominated transport.
A particular aspect of the electron-lattice coupling in
the manganites is the existence of a singly occupied e g
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
level in Mn3⫹ and its absence in Mn4⫹. The doubly degenerate e g level is split if the local symmetry is lower
than octahedral, and such a distortion will be expected
to occur, lowering the energy of the occupied state, but
at a cost in lattice energy. Such effects are generally
termed Jahn-Teller distortions (Ashcroft and Mermin,
1976). Indeed as we shall see, the crystal structure of the
parent compound LaMnO3 is orthorhombic as a consequence of a collective distortion of the structure and
gives rise to a Jahn-Teller splitting of the e g orbitals by
1.5 eV (Satpathy et al., 1996b). An Mn4⫹ site, however,
does not contribute to this lattice distortion, and should
be ‘‘more cubic’’ than its Mn3⫹ neighbors. It is generally
agreed that the net energy gain due to the Jahn-Teller
distortion can bind a charge carrier to its site, creating a
small polaron. Many authors refer to the charge carrier
in the activated regime as a ‘‘Jahn-Teller polaron.’’
A major motivation for the study of the manganites is
the richness of the low-temperature phases that emerge
upon doping. The end members of the series, RMnO3
and DMnO3 , where R is a rare-earth atom and D is a
divalent substituent such as Ca or Sr, are antiferromagnetic insulators. Mixtures of the two exhibit various
magnetic and charge-ordered ground states, which are
reviewed in this section.
A. Parent compounds
Two of the end-members of the series, LaMnO3 and
CaMnO3 , were studied in some detail by Wollan and
Koehler (1955) more than 40 years ago. The subject has
recently been revisited by Huang et al. (1997) in a detailed neutron scattering study of LaMnO3 . The x-raydiffraction work of Yakel (1955) showed that stoichiometric LaMnO3 is orthorhombic, belonging to space
group Pnma, while CaMnO3 is cubic, space group
Pm3m. The magnetic ground state of LaMnO3 is antiferromagnetic and was labeled A type by Wollan and
Koehler. As confirmed by later workers, the magnetic
moments lie in the a-c plane and are ferromagnetically
aligned along the a axis. Successive planes along b are
aligned antiparallel, as sketched in the inset of Fig. 4(a).
In this orthorhombic setting of the crystal, the a and c
axes are along Mn-Mn directions, with a⫽11.439 Å and
c⫽11.072 Å (Yakel, 1955). It is also convenient to use a
monoclinic cell with a m ⫽c m ⫽7.960 Å because of its
close correspondence to the underlying cubic perovskite
cell. CaMnO3 also orders antiferromagnetically, and may
be considered as two interpenetrating face-centered lattices with opposite spin. This C-type order, in the language of Wollan and Koehler, is sketched in Fig. 4(b).
The magnetic phases of LaMnO3 and CaMnO3 were
explained by Goodenough (1955) in terms of covalent
bonding between oxygen and Mn3⫹ and Mn4⫹ ions, respectively. In the case of LaMnO3 , the degeneracy of
the single e g orbital favors a cooperative Jahn-Teller dis-
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 7. Orbital structures at x⫽0 as a function of the antiferromagnetic interaction J S between the t 2g spins. The energy
parameters are chosen to be ˜␣ ⫽70 and ˜␤ ⫽2.5 [case (A)]. The
lower sketches demonstrate how the x 2 ⫺z 2 and y 2 ⫺z 2 orbitals are related to more common representations of d orbitals.
From Maezono et al., 1998b.
tortion (Kanamori, 1960), and the appearance of orbital
order. The issue of orbital and spin ordering has been
examined more recently by Maezono et al. (1998b). For
the undoped case (all sites occupied by Mn3⫹ ions), Maezono et al. obtain a mean-field phase diagram for spin/
orbital ordering as a function of the antiferromagnetic
interaction J s between t 2g core spins. The sequence of
phases is sketched in Fig. 7. For intermediate values of
J s the A-type antiferromagnetic phase is stabilized with
what is termed G-type orbital order. Linear combinations of e g orbitals in this state are in the form d z 2 ⫺x 2
and d z 2 ⫺y 2 , with lobes along Mn-O-Mn directions; i.e.,
the diagonals of the orthorhombic basal plane. The orbital state alternates between neighboring Mn3⫹ ions;
the relative orientation of the spin and orbital order are
shown in Fig. 8. Inoue and Maekawa (1995) have suggested, recalling earlier work by de Gennes, that the
ground state at low doping would exhibit spiral magnetic
order. Mishra et al. (1997), to the contrary, have demonstrated that canted (and presumably spiral) states depend on electronic parameters and are not necessarily
the ground state of a double exchange model. More recently, an ab initio density-functional calculation (Popovic and Satpathy, 2000) has shown that the orbital and
structural order arise simultaneously, driven by bandstructure energy contributions.
Detection of orbital ordering requires sensitivity to
the asphericity of the electronic charge density in the
orbitally ordered state, which leads to intensity at Bragg
peaks indexed as (h00) and (0k0), with h,k odd, that
are nominally extinct. In this case, Murakami, Hill et al.
(1998) observed resonant intensity at the (300) reflection, peaking at an x-ray energy 3 eV above the manganese K absorption edge. The resonant signal at the K
edge arises from splitting ⌬ among unoccupied 4p levels
via, perhaps, Coulomb interaction between 4p and polarized 3d levels, as sketched in Fig. 8. With the incident
radiation polarized normal to the scattering plane (␴)
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 8. Schematic view of the orbital and spin ordering in the
a-b plane of the perovskite manganite LaMnO3. This is the
orbital G spin A state shown in Fig. 7. The schematic energylevel diagram of Mn 4p x,y,z in the orbitally ordered state is
shown below. From Murakami, Hill et al., 1998.
and with frequency ␻, the cross section is zero unless the
scattered x rays are polarized (␲) in the scattering plane,
in which case
I ␴res, ␲ ⬇
A 4 sin2 ␺
兵 1⫹4 关共 ␻ ⫺ ␻ 0 兲 /⌫ 兴 2 其 2 ⌫ 2
Here ␻ 0 is the resonant energy and ⌫ ⫺1 is the lifetime of
the excited state. The angle ␺ is zero when the crystal c
FIG. 9. Azimuthal-angle dependence of the intensity of the
orbital ordering reflection (3, 0, 0) normalized by the fundamental reflections (2,0,0) and (4,0,0) at E⫽6.555 keV and at
room temperature. The solid curve is the calculated intensity.
Inset: Schematic view of the experimental configuration and
definition of the polarization directions. From Murakami, Hill
et al., 1998.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 10. Structures of the n⫽1, n⫽2, and n
⫽⬁ members of the Ruddlesden-Popper series of lanthanum manganites. From Moritomo et al., 1996.
axis is normal to the scattering plane and 90° when in
the scattering plane. A key test that the observed intensity arises from orbital ordering is found in the dependence of the (300) intensity on ␺, as shown in Fig. 9.
Some questions (Benfatto et al., 1999) have been raised
as to whether the observed signal is directly attributable
to orbital ordering or to Jahn-Teller distortions of the
same symmetry.
Materials of the type AMnO3 are the end (n⫽⬁)
members of so-called Ruddlesden-Popper series
A n⫹1 Mnn O3n⫹1 . The n⫽2 members of the series also
show dramatic magnetoresistive effects, and have been
widely studied. This compound has space group
I4/mmm and, as seen in Fig. 10, consists of square MnO
planes arranged in pairs, spaced from each other by AO
planes, and from adjacent pairs, by double AO planes.
The ‘‘parent’’ compound in this series, corresponding to
all manganese atoms nominally in the Mn3⫹ state, is
R 2 SrMn2O7. Almost all work reported has been done
R 2⫺2x Sr1⫹2x Mn2O7 with x⭓0.3. These will be discussed
in the next section.
the b and c axes in the parent compound interchanged.
The magnetic order is antiferromagnetic of the chargeexchange (CE) type described by Wollan and Koehler
(1955). A sketch of the complex orbital, charge, and spin
order is shown (Tokura and Tomioka, 1999) in Fig. 11.
While the spins are purely antiferromagnetic, the charge
and orbital ordering occurs in alternate b-c planes, contrary to expectations for large V NN , giving rise to
charge-stacked order. A Monte Carlo simulation carried
out recently (Hotta et al., 1999) considered Hund’s rule
J H , Jahn-Teller ␭, and antiferromagnetic J AF interactions. Using either a 4⫻4⫻2 bilayer or a 4⫻4⫻4 cube,
they find that the observed c axis charge-stacked state
has lowest energy at intermediate values of J AF so long
as V NN does not exceed ⯝0.2 eV. The bare value V NN
⯝3.6 eV is reduced below this critical value by the large
dielectric constant of the manganites. Stabilization of
this state requires the assistance of cooperative JahnTeller phonons. At larger and smaller values of J AF , the
B. Doped compounds
1. Charge ordering at half filling: n ⫽⬁
Upon doping, the manganites exhibit a wide variety of
ordered states, including ferromagnetic and chargeordered phases, in addition to the antiferromagnetism
and orbital ordering described above. In those compounds in which the proportions of Mn3⫹ and Mn4⫹ ions
are rational fractions, charge- and orbital-ordering effects are particularly pronounced. Charge ordering
might be expected as a consequence of nearest-neighbor
Coulomb repulsion V NN . However, the observed ordered structure in the 3D materials suggests a more
complicated explanation. The first clear evidence for
charge ordering was found in La0.5Ca0.5MnO3 by Chen
and Cheong using electron microscopy (Chen and
Cheong, 1996). Electron-diffraction images exhibit commensurate superlattice peaks at low temperature, consistent with alternation of Mn3⫹ and Mn4⫹, as expected for
dominance of Coulomb repulsion. The crystal structure
of this phase can be indexed in space group Pbnm, with
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 11. Spin, charge, and orbital ordering pattern of the CE
antiferromagnetic type observed for most of the x⬇1/2 manganites. The e g -orbital ordering on Mn3⫹ sites is also shown.
The Mn4⫹ sites are indicated by closed circles. Note that Mn3⫹
sites form chains along a, contrary to expectations from Coulomb considerations. From Tokura and Tomioka, 1999.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 12. Charge/orbital-ordered phase diagrams of various
RE1/2Sr1/2MnO3 and RE1/2Ca1/2MnO3 (RE⫽Pr, Nd, and Sm)
plotted in the magnetic field-temperature plane. The phase
boundaries have been determined by measurements of the
magnetic-field dependence of resistivity and magnetization at
fixed temperatures. Low-temperature data were obtained utilizing pulsed magnetic fields up to 40 T. From Tokura and
Tomioka, 1999.
stable states of the system are G-type antiferromagnetic
(neighboring Mn spins antiparallel) and ferromagnetic
states, respectively. This close relationship among various phases is observed experimentally.
Substitution of Nd for La narrows the bandwidth and
destabilizes the ferromagnetic state. Nd0.5Sr0.5MnO3 exhibits a narrow CE-type, charge ordered state (Kajimoto
et al., 1999), while Pr0.5Sr0.5MnO3 , with a still narrower
band exhibits A-type antiferromagnetic order and remains conductive (Kawano et al., 1997). Indicative of the
delicate balance among charge, spin, and orbital ordering, Pr0.5Ca0.5MnO3 is reported to order in the chargeordered, CE antiferromagnetic phase (Tokura and Tomioka, 1999). Another manifestation of this balance is
found in the field-induced melting of the charge ordered
state. This was first reported by Kuwahara et al. (1995)
for Nd0.5Sr0.5MnO3 and subsequently to occur in Pr-Ca,
Nd-Ca (Tokunaga et al., 1998), Pr-Sr (Tomioka et al.,
1995), and Sm-Ca analogs, as seen in Fig. 12. The transition is, as seen, highly hysteretic and accompanied by
drop in resistivity by as much as four orders of magni-
tude at low temperatures, as seen in Fig. 13. The sequence of phases with various combinations of trivalent
and divalent ions in equal proportion has been reviewed
in detail recently by Tokura and Tomioka (1999).
A related compound, La0.5Sr1.5MnO4 , similar to the
214 family of cuprate superconductors and having equal
numbers of Mn3⫹ and Mn4⫹ ions, was found by Moritomo et al. (1995) and Bao et al. (1996), using transport
and electron diffraction, to exhibit charge and magnetic
order. These results were confirmed by neutron diffraction (Sternlieb et al., 1996) and by x-ray diffraction at
the (3/2, 3/2, 0) superlattice position, making use of
anomalous dispersion terms for Mn3⫹ and Mn4⫹ ions at
the K edge (Murakami et al., 1998). The latter result was
part of a path-breaking experiment by Murakami et al.
in which orbital ordering, which doubles the charge ordered cell, was detected. The combined magnetic, orbital, and charge structure is shown in Fig. 14. Due to
the anisotropy of the e g wave functions, the atomic scattering factors must be treated as tensors near an absorption edge. These give measurable intensity at the K edge
at the otherwise forbidden (3/4, 3/4, 0) position. The intensity varies with azimuthal angle as in La0.5Ca0.5MnO3
and shows the required change in polarization. This
work represented one of the first direct observations of
orbital ordering, although earlier reports of (1/4, 1/4, 0)
reflections in electron diffraction (Moritomo et al., 1995;
Bao et al., 1996) may also have been due to orbital ordering.
2. Charge ordering at half filling: n ⫽2
In the n⫽2 bilayer compounds, equal numbers of
Mn3⫹ and Mn4⫹ ions are expected for the composition
RSr2Mn2O7. Here, however, charge ordering is less
prevalent. Using electron diffraction, Li et al. (1997) and
Kimura et al. (1998) detected the presence of charge order in LaSr2Mn2O7 at 110 K via an increase in resistivity
and the appearance of superlattice peaks in x-ray diffraction. The additional peaks have the wave vector q
⫽(1/4,1/4,0), and are consistent with charge/orbital ordering. However, recent neutron-diffraction results
(Kubota, Fujioka et al., 1999) find that it does not persist
FIG. 13. Changes in the resistivity of a Nd1/2Sr1/2MnO3 crystal after cooling to T⫽2.49 K in zero field. The resistivity shows jumps
with increasing and decreasing fields at lower and upper critical fields, respectively, as indicated by crosses. From Kuwahara et al.,
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
M. B. Salamon and M. Jaime: Manganites: Structure and transport
tution of La for Nd reduces this lower Néel temperature,
leaving antiferromagnetic order of the Mn ions in which
the Mn bilayers form ferromagnetic sheets with the moments aligned and antialigned with Mn-O-Mn bond directions. Below the Néel temperature T N ⬇150 K of the
Mn spins in NdSr2Mn2O7 , Maezono et al. (2000) predicted ordering of d x 2 ⫺y 2 orbitals concurrent with
A-type antiferromagnetic order. A remarkable study by
Takata et al. (1999) combined Rietveld analysis of
powder-diffraction data with the maximum-entropy
method. In this method, all deformations of the electron
density allowed by symmetry are considered, and result
in electron density maps which, as shown in Fig. 15,
clearly demonstrate the ordering of orbitals into the expected d x 2 ⫺y 2 configuration.
3. Other fractional doping levels
FIG. 14. Schematic view of the charge, spin, and orbital ordering in a layered perovskite manganite, La0.5Sr1.5MnO4. The
stacking vector along the c axis, shown in the figure, means
that the second layer is shifted by one unit cell in this direction.
From Murakami, Kawada et al., 1998.
at 80 K. They indicate that the magnetic order is predominantly A type, with some admixture of CE antiferromagnetic order and suggest a competition between
d x 2 ⫺y 2
d 3x 2 ⫺r 2 /d 3y 2 ⫺r 2 charge order. This result is not surprising, as the Monte Carlo results (Hotta et al., 1999) for
bilayers suggest that the charge ordered state is stable
only over a very narrow part of the phase diagram without cooperative Jahn-Teller distortions. The series from
R⫽Nd to R⫽La has been studied in some detail by
Moritomo and co-workers (1999). At low temperatures
(T⭐20 K) the Nd compound has been found to order in
a complicated antiferromagnetic state, in which both Mn
and Nd moments are canted (Battle et al., 1996). Substi-
It is natural to expect that charge ordering phenomena might appear at other rational values of doping
level. Indeed, Cheong and Chen (1998) have suggested
that carrier concentrations of x⫽N/8 per Mn atom characterize special points on the phase diagram. It is somewhat difficult to make this case apart from the x⫽4/8
materials discussed in the previous section. There appears to be a boundary between CE- and C-type antiferromagnetic order in the La1⫺x Cax MnO3 system near
x⫽7/8. When Mn3⫹ and Mn4⫹ are in the ratio 1:2, as
they are in La1/3Ca2/3MnO3 , the system was thought to
order into diagonal charge stripes (Cheong and Hwang,
2000). The stripe order is closely related to the x⫽1/2
charge ordering, with an extra stripe of Mn4⫹ inserted
between diagonal Mn3⫹-Mn4⫹ rows. The original interpretation, based on electron diffraction and termed the
bistripe mode, has been shown to be inconsistent with
more recent x-ray and neutron data (Radaelli et al.,
1999). The two structures are compared in Fig. 16.
In many of these materials, a ferromagnetic phase appears in a range of compositions, optimized for a wide
range of materials when the Mn4⫹/Mn3⫹ ions are in the
FIG. 15. Contours containing a
charge density of 0.4 e Å⫺3 , obtained by maximum-entropy
analysis of synchrotron-x-ray
powder-diffraction data for
NdSr2Mn2O7 at 19 K, showing
the MnO6 double layers. The
schematic orbital structure is
also shown. From Takata et al.,
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 16. Schematic diagram of
charge ordering patterns in
La1/3Ca2/3MnO3 showing alternating chains of Mn4⫹ (open
circles) and Mn3⫹ (orbitals)
separated by double chains of
Mn4⫹ sites (bistripe model).
The Wigner-crystal model is in
better agreement with observations. From Radaelli et al.,
ratio 3:5 (x⫽3/8) (Cheong and Hwang, 2000). It is, of
course, this range of compositions that is of the most
interest, as these are the most metallic compounds. We
defer a discussion of the low-temperature properties of
the ferromagnetic composition range to the next section.
Figure 17 shows the sequence of phases for
Nd1⫺x Srx MnO3. The sequence A-type antiferromagnetic to ferromagnetic to A-type antiferromagnetic to
C-type antiferromagnetic is typical of all the 3D perovskites. The intermediate CE phase at x⫽1/2 is not universally present. The surrounding diagrams in the figure
sketch the orbital ordering deduced from the x-ray studies described above.
The n⫽2 compounds have been studied over a narrower composition range, but also show a variety of lowtemperature phases (Kubota, Fujioka et al., 1999; Argy-
FIG. 17. Phase diagram of Nd1⫺x Srx MnO3 crystals. The
sketches illustrate the orbital structure types associated with
each phase. The letters within the phase diagram refer to paramagnetic (P), ferromagnetic (F), A-type antiferromagnetic
(A), CE-type antiferromagnetic (CE), and C-type antiferromagnetic (C) spin ordering. From Okuda et al., 1999.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
riou et al., 1999). In La2⫺2x Sr1⫹2x Mn2O7, the
ferromagnetic order below x⫽0.32 has all spins normal
to the double MnO planes, designated in Fig. 18 as FMII. Between x⫽0.3 and x⫽0.4, the spins lie in the plane
(FM-I). Beyond x⫽0.4, there is canting, or else an admixture of FM-I and AFM-II, with spins in the double
planes parallel, but antiparallel stacking of successive
double planes.
4. Theoretical situation
Since Goodenough’s early work (Goodenough, 1955)
on covalence effects, the sequence of phases as a function of doping has remained a topic of considerable interest. Semicovalency, a term coined by Goodenough,
arises when the overlap of spin-polarized sp orbitals of
manganese ions with unoccupied orbitals of the oxygen
allow only covalent bonds involving electrons of one
spin direction. Four possible Mn-O-Mn configurations
La2⫺2x Sr1⫹2x Mn2O7 (0.30⭐x⭐0.50). AFM-I (AFM-II) indicates the planar A-type AFM structure with AFM (FM) intrabilayer coupling and FM (AFM) interbilayer coupling. FM-I
and FM-II stand for the ferromagnetic structures with spin
within the ab plane and along the c axis, respectively. As for
x⫽0.50, only AFM-I exists in phase I, while AFM-I and CEtype AFM coexist in phase II. From Kubota, Fujioka et al.,
M. B. Salamon and M. Jaime: Manganites: Structure and transport
TABLE I. The sequence of phases as the fraction x of divalent substituents is increased. The magnetic and orbital states are indicated (Goodenough, 1955).
x range
Magnetic order
A-type antiferromagnetic
CE-type antiferromagnetic
C-type antiferromagnetic
G-type antiferromagnetic
are possible: antiferromagnetic if both empty Mn bonds
point toward the bridging oxygen; ferromagnetic if one
such bond connects to the oxygen, the other Mn-O bond
being essentially ionic; paramagnetic in the absence of
covalency; and ferromagnetic and conducting if there is
disorder that allows double exchange coupling. In this
scheme, the end points can be explained, as noted
above. At x⫽1 all six Mn-O-Mn bonds are antiferromagnetic giving rise to G-type order, while at x⫽0, only
four bonds can be covalent, leading to A-type antiferromagnetic order. Table I summarizes the sequence of
phases predicted by Goodenough. At intermediate concentrations, there is an admixture of adjacent phases.
A more recent study of the doping phase diagram has
been carried out by Maezono et al. (1998a, 1998b), using
a double exchange model in which the orbital degeneracy of the e g levels is included by means of an isospin
ជi . The mean-field energy associated with orbital order
ជi 典 T
ជi , which is solved along with
has the form ⫺ ␤ 兺 i 具 T
spin ordering. For ␤ ⫽0 and reasonable values of the
underlying antiferromagnetic superexchange interaction
J s , the ferromagnetic state is stable from x⫽0 to x
⬇0.6 at T⫽0. When ␤ is sufficiently large, the model
predicts the progression A→F→A→C→G. The CE
phase is not found. The phase diagram, along with spin/
orbital ordering patterns, is shown in Fig. 19. The model
leading to Fig. 19 does not include Jahn-Teller coupling,
and predicts, as seen, that the orbital state at x⫽0 is
G-type ordering of y 2 ⫺z 2 and z 2 ⫺x 2 orbitals. This dif-
FIG. 19. Free energies for each spin alignment as a function of
the antiferromagnetic interaction J S between t 2g spins at x
⫽0. The energy parameters are chosen to be ˜␣ ⫽70 and ˜␤
⫽2.5 [case (A)]. These correspond to the orbital structures
shown in Fig. 7. From Maezono et al., 1998b.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
Orbital order
normal to ferromagnetic sheets
charge ordered
fers from the experimentally deduced model shown in
Fig. 8. Maezono et al. considered the addition of JahnTeller coupling and find that it tends to suppress the
ferromagnetic spin arrangement and to favor the 3x 2
⫺r 2 /3y 2 ⫺r 2 ordering found experimentally. The calculated ferromagnetic region is less robust than is observed; cf. Fig. 17. Maezono et al. suggest that increasing
the bandwidth ⬀t 0 and the antiferromagnetic exchange
J s ⬀t o2 , while keeping the other parameters constant,
moves the ferromagnetic/antiferromagnetic boundary to
larger values of x. Maezono et al. (2000) have also extended their calculations to the layered manganites, including orbital ordering, spin canting, and c-axis lattice
deformation. The results are in qualitative agreement
with experiment. This model includes on-site Coulomb
interactions, but ignores intrasite repulsion. A calculation by Mishra et al. (1997) suggests such terms favor the
antiferromagnetic phase near x⫽1/2.
A quite different approach has been taken by Yunoki
and co-workers (Yunoki, Hu et al., 1998; Yunoki, Moreo
et al., 1998) via Monte Carlo simulation. The model includes two e g orbitals, the usual Kondo Hamiltonian for
the double exchange model, and Jahn-Teller coupling to
classical phonons. The core spins are assumed to be classical. In chains up to 18 sites in length, the carrier density is found to be a discontinuous function of the chemical potential, suggesting phase separation. Extended to
clusters of 4 2 sites, the system still exhibits phase separation. Above a critical value of the Jahn-Teller coupling
␭, there is separation between a hole-rich ferromagnetic
phase and hole-poor antiferromagnet, as shown in Fig.
20; note that 具 n 典 here is 1⫺x in our notation. This
model does not include long-range Coulomb interactions which should prevent macroscopic separation between doped-in holes and their Ca2⫹ or Sr2⫹ donor sites.
This point is discussed in a review article by Moreo et al.
(1999), where it is suggested that either droplets or polarons may be the most likely configuration. It is not
clear the degree to which these 2D Monte Carlo results
carry over to 3D manganites.
Density-functional methods have been applied to
LaMnO3 and CaMnO3 by Satpathy et al. (1996a, 1996b).
For LaMnO3 the splitting between t 2g and singly occupied e g levels is found to be 2.0 eV, with a 1.5 eV JahnTeller splitting between d 3z 2 ⫺r 2 (lower) and d x 2 ⫺y 2 orbitals. Both the experimental results and the calculations
of Maezono suggest that a linear combination of these is
required in the ground state. The bandwidth is found to
M. B. Salamon and M. Jaime: Manganites: Structure and transport
ones. At low temperatures, the metallic state is characterized by almost fully spin-polarized bands (Park et al.,
1998) and a relatively high residual electrical resistivity
in the range 50–80 ␮⍀ cm.
1. Magnetic properties
FIG. 20. Monte Carlo simulations of double exchange models
with scaled Jahn-Teller interaction parameter ␭: (a) Charge
per site 具 n 典 vs chemical potential ␮ in the limit that the Hunds’
exchange J H ⫽⬁, using ␭⫽1.5, antiferromagnetic coupling J ⬘
⫽0.05; (b) ␮ at ␭⫽1.2 using a 42-site cluster and at T⫽1/50.
The two sets of points that produce the hysteresis loops are
obtained by increasing and decreasing ␮ using ⬃104 sweeps at
each ␮; (c) phase diagram of the 2D model at J H ⫽8.0, J ⬘
⫽0.05. Jumps that appear in (b) are signatures of phase separation (PS). The phases are denoted S-F and S-AF (FM- and
AFM-spin order, respectively) and O-D, O-F, and O-AF (disordered, uniform, and staggered orbital order, respectively).
From Yunoki, Moreo, and Dogotto, 1998.
be approximately 1.5 eV, with a small gap at the Fermi
energy. On doping, the 3z 2 ⫺r 2 band is emptied, leading
to the metallic state. Similar results have been reported
by Pickett and Singh (1996, 1997a, 1997b).
There remain doubts as to which aspects—double exchange, Jahn-Teller effects, polarons, phase separation,
among others—are essential to produce a minimal theoretical picture of the manganites. The different contributions are discussed in a recent survey by Loktev and
Pogorelov in which they search for common elements
needed to produce a strongly field-sensitive metalinsulator transition (Loktov and Pogoretov, 2000). They
conclude that the coexistence of various spatial scales, a
strong competition between magnetic, orbital, and lattice ordering, and the tendency toward the formation of
domains with sharply contrasting properties must remain central factors in any eventual theory.
C. Ferromagnetic regime, 3D materials
By far, the most attention has been directed to those
compositions that have a low-temperature, metallic, ferromagnetic state. Such a phase is a common feature in
the phase diagram of the mixed manganites
A 1⫺x A x⬘ MnO3, where (A, A ⬘ )⫽(La, Ca), (La, Sr), (La,
Ba), (La, Pb), (Sm, Sr), and (Nd, Sr) to mention some of
them, for values of x close to 3/8 and extending from 0.2
to 0.5. The metallic state was studied to some extent in
the past but not until recently were the intrinsic properties identified and separated from sample-dependent
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
Perhaps the simplest question about the ferromagnetic regime in manganites, and one we should be able
to answer, is to what extent are manganites standard
Heisenberg-like ferromagnets? Magnetization vs magnetic field results in films of (La, Ca)MnO3 reveal typical
hysteresis loops with coercivity H c of about 30–50 Oe
and a saturation field H S close to 1/2 T (McCormack
et al., 1994). The saturation magnetization at T⬇0
matches well the spin only value expected from
all 3d electrons present in manganese ions:
M S ⫽x⫻Mn3⫹(S⫽4/2)⫹(1⫺x)⫻Mn4⫹(S⫽3/2)⫽4x ␮ B
⫹3(1⫺x) ␮ B for concentrations x⬇0.3. When doping is
lower than 0.2 or larger than 0.5, the saturation magnetization vanishes very quickly, Fig. 2 (Jonker and van
Santen, 1950). Measurements in La0.67(Pb, Ca)0.33MnO3
single crystals performed by the authors (Jaime et al.,
1998) reveal that the magnetization decreases when
the temperature is increased as expected for spin-wave
excitations, i.e., M(T)⫽M(0)⫺BT 3/2⫺CT 5/2 ¯ where
B⫽0.0587g ␮ B (k B /D) 3/2 and D is the spin-wave stiffness constant. At temperatures one order of magnitude
lower than the Curie temperature the T 3/2 term dominates the temperature dependence of M, but at T C /2
higher power terms take over, and at T C the magnetization vanishes abruptly deviating from Heisenberg-like
behavior, similar to data obtained in Ca-free single crystals [Fig. 5(a)]. The stiffness constant determined from
the magnetization measured in a field of 10 kG is D
⫽165 meV Å2 in very good agreement with neutrondiffraction data.
Neutron-diffraction studies in the manganites started
with the work by Wollan and Koehler (1955) who measured La1⫺x Cax MnO3 for various compositions. They
found pure ferromagnetic behavior only in the narrow
composition range 0.25⬍x⬍0.4, confirming Jonker’s results on the composition dependent magnitude of the
saturation moments. They did not study the temperature
dependence in detail. During the last five years,
neutron-scattering techniques were exploited to study
the manganites in greater detail, and proved to be both
extremely interesting and highly nontrivial. At low temperatures and in all compositions, the magnetic inelastic
spectra, i.e., energy scans at constant momentum transfer Q, show typically ferromagnetic spin-wave peaks
with essentially zero gap and a quadratic dependence on
Q at small Q (Lynn et al., 1996; Fernandez-Baca et al.,
1998). Well below T C the spin-wave excitations decrease
in energy and increase in amplitude when the temperature is increased as expected, Fig. 21. However, as the
temperature approaches T C , a new feature develops in
the spectrum, consisting of a zero energy quasielastic
component also known as the central component. The
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 22. Inelastic neutron-scattering spectra at 250 K (T C )
and Q⫽0.09 Å⫺1 as a function of applied field H. At this temperature there are still spin waves present, but the spectrum is
dominated by the quasielastic scattering. In a field of 2 T, the
spin-wave signal increases in intensity and sharpens, while the
quasielastic response decreases. At 6 T, the spin-wave peaks
have moved out of the range of the energy scan. From Lynn
et al., 1997.
FIG. 21. Magnetic inelastic neutron scattering below T C (a)
La1⫺x Cax MnO3 for x⫽1/3 at a momentum transfer
Q⫽0.07 Å⫺1. At 200 K the spectrum is dominated by the
spin waves in both energy gain and loss. However, as T
→T C , a quasielastic (near E⫽0) component develops
and grows to dominate the spectrum (Lynn et al., 1997). (b)
Similar energy scans for Pr2/3Sr1/3MnO3 (PSMO) and
Nd2/3Sr1/3MnO3 (NSMO) at momentum transfer q⫽0.08 (in
reciprocal-lattice units). From Fernandez-Baca et al., 1998.
central component, more notorious in those compositions with lower T C , persists at least up to 1.25T C with
little T dependence above T C and follows the quadratic
dependence in Q expected for spin diffusion. The magnetic nature of the central mode has been verified with
measurements in magnetic fields. As the magnetic field
is increased at constant temperature, the central mode is
suppressed and the spin-wave components sharpen as
they increase in energy and intensity, Fig. 22 (Lynn et al.,
1997). The temperature dependence of the quasielastic
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
peak is anomalous. For typical isotropic ferromagnets,
such as Ni, Co, Fe any quasielastic scattering below T C
is too weak to be observed directly in the raw data
(Vasiliu-Doloc et al., 1998), while in some manganites
this feature starts to dominate at temperatures well below T C . These results together suggest a correlation between two mechanisms (spin waves on one side and
quasielastic neutron-scattering centers, possibly small
polarons, on the other) that dominate different temperature, ranges (low and high temperatures, respectively)
and coexist in an extended region around T C . We shall
discuss both the high-temperature regime and the coexistence region in detail later.
The energy dispersion for spin waves has also been
measured in detail, and from it the spin-wave stiffness
constant was calculated by fitting the low-Q region to
the expression E⫽⌬⫹D(T)Q 2 for a number of compounds (Fernandez-Baca et al., 1998). Neutrons do not
detect a measurable energy gap ⌬ in the spin-wave dispersion relation; based on the experimental errors an
upper limit has been estimated not to exceed 0.05 meV,
which indicates that manganites are soft isotropic ferromagnets, comparable to very soft amorphous ferromagnets. As the temperature is increased and the spin-wave
energy renormalizes, the stiffness constant decreases following a law expected for two-magnon processes (Mattis, 1981),
D 共 T 兲 ⫽D 共 0 兲共 1⫺AT 5/2兲 ,
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 23. Spin-wave stiffness coefficient D in E⫽E 0 ⫹Dq 2 as a function of temperature: (a) in La0.8Sr0.2MnO3; (b) in
La0.7Sr0.3MnO3. (c) in La0.67Ca0.33MnO3 (d) in Nd0.7Sr0.3MnO3 (NSMO) and Pr0.63Sr0.37MnO3 (PSMO). In (a) and (b), the solid
curves are fits to the usual Bloch law. D appears to vanish at the ferromagnetic transition temperature, as expected for a
conventional ferromagnet; the dashed curves are fits to a power law in (1⫺T/T C ). In (c) D does not vanish at the ferromagnetic
transition temperature, in contrast to the behavior of conventional ferromagnets (Lynn et al., 1996). In (d) the spin-wave stiffness
coefficient vs T/T C vanishes at T C for PSMO (open circles), but not for NSMO (solid circles). The solid line is the fit to the
mode-mode coupling and hydrodynamic theories at high temperatures. The dashed line is an extrapolation to T⫽0 from the
low-temperature mode-mode coupling theory. From Fernandez-Baca et al., 1998.
A⫽ 共 ␷ 0 l 2 ␲ /S 兲共 k B /4␲ D 0 兲 5/2␨ 共 25 兲 ,
␷ 0 is the unit-cell volume, S is the manganese spin, ␨ ( 25 )
is the Riemann zeta integral, and l 2 is a parameter which
gives the mean-square range of the exchange interaction, usually comparable to the square of the lattice parameter a 2 . As a rule the stiffness constant decreases
when the temperature is increased, as predicted by this
formalism, only at low temperatures, and deviates from
it as the temperature reaches T C . The temperature dependence in the transition region differs for LaSr and
LaCa compounds. For LaSr it is well described by an
expression of the form D(T)⬃ 关 (T⫺T C )/T C 兴 ␯ ⫺ ␤ with
the critical exponent ( ␯ ⫺ ␤ ) very close to the value observed in 3D ferromagnets like iron, cobalt, and nickel
(Vasiliu-Doloc et al., 1998). For LaCa compounds, the
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
stiffness constant seems to remain finite at T C , a result
that clearly goes beyond standard ferromagnetism, as
seen in Fig. 23.
Computation of the spin-correlation length from neutron measurements of the static wave-vector-dependent
susceptibility reveals that while the correlation length
diverges at T C ⫽300.9 K for Pr0.63Sr0.37MnO3, it remains
finite and around 20 Å down to 0.95T C ⫽0.95⫻197.9 K
for Nd0.7Sr0.3MnO3. It has been proposed (FernandezBaca et al., 1998) that magnetism alone cannot explain
the exotic spin dynamical properties of these systems
and that the increased electron-lattice coupling plays a
role. Support for these claims is found in the energy
dispersion relation for spin waves. Indeed, when studied
in all the way to the zone boundary an anomalous
softening/broadening is found in LaCa compounds while
the energy dispersion relation for LaSr compounds is
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 24. Magnon dispersion curves open (symbols) along
[0,0,␰] and [␰,␰,0] directions for PSMO, NSMO, and LCMO as
in Fig. 23 at 10 K. Superposed on these are optical-phonon
curves (solid symbols) for LCMO. The solid lines are a fit of a
nearest-neighbor spin-wave model to the data for ␰ ⬍0.1. The
spin waves become strongly damped and nondispersive once
their energy intersects the optical-phonon branches. From Dai,
Huang et al., 2000.
not anomalous. The softening and broadening cannot be
explained by the double exchange mechanism; in fact,
no correlation between the Curie temperatures and dispersion relation for LaCa, NdSr, and PrSr compounds is
found. On the other hand, a remarkable correlation with
optical-phonon modes, as shown in Fig. 24, strongly suggests that magnetoelastic coupling (possibly between
magnons and Jahn Teller modes) is responsible for the
softening (Dai et al., 2000). A theoretical study shows
that in fact phonon-magnon interactions can result in
magnon broadening in low-T C (narrow bandwidth)
manganites provided a large enough magnon-phonon interaction is present (Furukawa, 2000), an interaction
that could at least in principle originate in the fact that
the metallic state in these compounds is in close proximity to the charge ordered insulating state. In conclusion,
La-Sr, and perhaps Pr-Sr, manganite can be well described as a Heisenberg ferromagnet, but other manganites cannot. It may be that in those compounds with
smaller tolerance factors and correspondingly lower
transition temperatures, large phonon-magnon interactions and Jahn-Teller modes begin to play an important
2. Electrical transport properties
The low-temperature transport properties of manganites have been studied in some detail only in recent
years. Those properties, which require a finite electric
current through the sample, have proven to be sensitive
to extrinsic effects as grain boundaries and magnetic domain boundaries. These effects are related to grain
boundary charge/spin tunneling-limited transport and
have attracted a great deal of attention because of the
potential applications of manganites as spin-valve-like
devices. A large low-field/low-temperature magnetoreRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 25. Two different treatments of the dependence of the
resistivity ␳ (H,T) on the magnetization M(H,T): (a) an exponential dependence in La0.7Ca0.3MnO3, using data taken at
H⫽10, 20, 30, 40, and 50 kOe that progress, respectively, from
low to high M values; solid line, a least-squares fit to the data
(Hundley et al., 1995); (b) similar analysis for a La1⫺x Srx MnO3
crystal (x⫽0.175). The points are obtained from the ␳ (B) and
M(B) curves at respective temperatures. The solid line is obtained using the ␳ (T) and M(T) data in a field of 0.5 T. The
behavior is quadratic at small values of the magnetization.
From Tokura and Tomioka, 1999.
sistance can be obtained when samples are polycrystals
prepared with conveniently small crystallographic grains
(Schiffer et al., 1995). This spin-tunneling limited magnetoresistance is based on good spin polarization of individual grains or magnetic layers in the system and vanishes as thermal entropy kills intragrain polarization,
more or less rapidly as practical temperatures are
reached (Hwang et al., 1996). Epitaxial films do not
show low-field magnetoresistance (Li, Gupta et al.,
1997). There is also a high-field magnetoresistance component associated with the grain boundaries that is essentially temperature independent between 5 and 280 K,
and is most likely related to alignment of spins in a magnetically disordered region near the grain boundaries.
The low- and high-field magnetoresistance, studied as
functions of the grain size by the group in Barcelona
M. B. Salamon and M. Jaime: Manganites: Structure and transport
(Balcells et al., 1998), show saturation in low fields and
monotonic increase in high fields for grain sizes equal to
or smaller than 1 ␮m, revealing the distinct origin of
both contributions. La2/3Sr1/3MnO3 samples, where grain
sizes are 30 nm or smaller, do not show a metallic electrical resistivity in any temperature range, not even in
the low-temperature ferromagnetic phase. This behavior
has been attributed to a Coulomb blockade contribution
to the resistivity, similar to that observed in such other
half metallic systems as CrO2 (Coey et al., 1998).
The intrinsic low-temperature transport properties of
ferromagnetic manganites are far from trivial, and in
fact have proven to be a challenge. The intrinsic magnetoresistance of these compounds vanishes at low temperatures, and correlates well with the magnetization,
i.e., saturated magnetization⫽null magnetoresistance. A
detailed study of these correlations indicates that the
resistivity and magnetization are related by the empirical expression ␳ (H,T)⫽ ␳ m exp关⫺M(H,T)/M0兴 over a
wide temperature range all the way up to the Curie temperature (Hundley et al., 1995), clear evidence of the
double exchange interaction at play [Fig. 25(a)]. However, in contradiction to the exponential fit, ␳ (H,T) is
found to decrease as the square of the magnetization
near T C (Tokura and Tomioka, 1999), as seen in Fig.
As a consequence of the double exchange mechanism,
charge carriers at low temperatures are spin-down holes
moving in an S⫽5/2 background. There are no propagating up-spin hole states in the S⫽2 manifold; they
only exist localized on sites at which the t 2g core is not
ferromagnetically aligned. Therefore single-magnon
scattering processes, which cause the resistivity of conventional ferromagnets to vary as T 2 , are suppressed.
Kubo and Ohata extended the standard perturbation
calculation of Mannari (1959) to consider two-magnon
processes, predicting a leading T 9/2 temperature dependence of the resistivity. However, a dominant T 2 contribution is universally observed in the manganites, and
has usually been ascribed to electron-electron scattering
(Urushibara et al., 1995; Schiffer et al., 1995; Snyder
et al., 1996). However, there is a general relationship between the coefficient A of the T 2 contribution to the
resistivity and the electronic heat-capacity coefficient ␥,
pointed out by Kadowaki and Woods (1986). In the
manganites, the ratio A/ ␥ 2 is more than an order of
magnitude larger (Jaime et al., 1998) than the
Kadowaki-Woods ratio for metals in which strong
electron-electron interactions are present. This suggests
that a different mechanism is involved. In a recent paper
(Jaime et al., 1998) resistivity data on single crystals of
composition La0.67(Ca, Pb)0.33MnO3 are discussed, demonstrating that the quadratic temperature dependence is
rapidly suppressed as the temperature is reduced. They
argue that the observed T 2 contribution reflects the reappearance of minority spin states that are accessible to
thermally excited magnons. Spin-polarized photoemission data, taken on films exhibiting square hysteresis
loops, indicate 100% spin polarization only at low temperatures, decreasing gradually as the temperature is inRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 26. Low-temperature behavior of the resistivity: (a) resistivity vs T 2 for magnetic fields up to 70 kG in a thin-film
sample of La2/3Ca1/3MnO3 . (b) Numerical derivative of the
data, ␣ H
⳵␳ / ⳵ T 2 ; 䊏, for zero field; 䊐, for 70 kG dashed curve,
the same quantity calculated for the magnon model with
Dq min /kB⫽44 K. Inset: raw data minus calculated residual resistivity (solid squares) and least-squares curve (dotted line) vs
temperature. The nonadditivity of the residual resistivity and
the quadratic component violate Matthiesson’s rule. From
Jaime et al., 1998.
creased (Park et al., 1998). Single crystals, which have
essentially no hysteresis, would be expected to depolarize more rapidly. Jaime et al. have extended Mannari’s
calculation to the situation in which a minimum magnon
energy is required to induce spin-flip transitions. At temperatures well below that energy, single magnon scattering is suppressed exponentially as predicted by Kubo
and Ohata. The treatment is in the context of the relaxation time approximation while a proper theory would
consider lifetime effects from magnon scattering using
Furukawa’s many-body approach. Nonetheless, the results are in qualitative agreement with the data. Bandstructure calculations (Pickett and Singh, 1996; Singh
and Pickett, 1998) also indicate that minority spin states
persist at E F , even at T⫽0 K.
Figure 26(a) shows the resistivity of a single-crystal
sample of La0.67(Ca, Pb)0.33MnO3 with T C ⫽300 K, vs the
M. B. Salamon and M. Jaime: Manganites: Structure and transport
square of the temperature in fields up to 70 kOe. The
data show a dominant T 2 temperature dependence with
evidence of a small T 5 contribution (10 ␮⍀ cm at 100
K). A calculation of the T 9/2 contribution predicted by
Kubo and Ohata for two-magnon processes predicts
only 0.5 ␮⍀ cm at 100 K with appropriate parameters. It
is likely, then, that this is the usual T 5 contribution from
electron-phonon processes. The inset to Fig. 26(b) shows
that the data do not follow a T 2 dependence to the lowest temperatures. Rather, they deviate gradually from
the curve ␳ 0 ⫹ ␣ (H)T 2 , fit over the range 60⭐T
⭐160 K, before saturating at an experimental residual
resistivity ␳ exp
0 ⫽91.4 ␮ ⍀ cm, comparable to values observed by Urushiba et al. (1995), but ⬃7% larger than
␳ 0 . This conclusion is not changed by including the T 5
contribution. Fits to data taken in various fields show
that ␣ H decreases with increasing field and is the source
of the small negative magnetoresistance at low temperatures. To quantify the disappearance of the T 2 contribution, Jaime et al. numerically differentiate the data, plotting ␣ (H) ⫺1 d ␳ /d(T 2 ) in Fig. 26(b). The T 5
contribution, not subtracted, gives a slight upward curvature to the data at higher temperatures.
The usual calculation of the electron-magnon resistivity (Mannari, 1959) has been extended to allow the
minority-spin subband to be shifted upward in energy
such that its Fermi momentum differs by an amount q min
from that of the majority subband. This should be a reasonable approximation in the intermediate temperature
regime in which both minority and majority bands have
substantial densities of states at E F . The one-magnon
contribution can then be written as ␳ ⑀ (T)⫽ ␣ ⑀ T 2 , where
␣ ⑀⫽
9 ␲ 3N 2J 2ប 5
8e 2 E F4 k F
冉 冊
m *D
I共 ⑀ 兲.
In this equation, NJ is the electron-magnon coupling
energy which is large and equal to ␮ ⫽W⫺E F in the
double exchange Hamiltonian of Kubo and Ohata; 2W
is the bandwidth. The magnon energy is given by Dq 2 ,
and we have defined
I共 ⑀ 兲⫽
sinh2 x
/2k B T, where Dq min
is the
The lower limit is ⑀ ⫽Dq min
minimum magnon energy that connects up- and downspin bands; this result reproduces Mannari’s calculation
in the limit ⑀ →0, and Kubo and Ohata’s exponential
cutoff for large ⑀. At high temperatures, the lower limit
of the integral in Eq. (10) can be set equal to zero, leaving only the coupling energy NJ⫽W⫺E F as a parameter. Equating the calculated value to the experimental
␣ fixes the coupling to be W⫺E F ⬇1.0 eV or W
⯝1.5 eV, in good agreement with a virtual crystal estimate of the bandwidth (Pickett and Singh, 1997a). Figure 26(b) shows the experimental data (1/␣ H ) ⳵␳ /dT 2 ,
and ␣ ⫺1
0 d( ␣ ⑀ T )/d(T ) assuming D(0)q min⫽4 meV
and including the temperature dependence observed experimentally, D(T)/D(0)⫽(1⫺T/T C ) 0.38 (FernandezBaca et al., 1998) which is important only at higher tem-
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
peratures. While the curve follows the data qualitatively,
it is clear that the minimum magnon energy is substantially larger than 4 meV at low temperatures, and decreases with increasing temperature.
This extension of the magnon resistivity calculation to
spin-split parabolic bands greatly oversimplifies the
changes in the minority-spin band that accompany magnetic ordering. As a consequence, the calculation cannot
be expected to represent accurately the cutoff of magnon scattering due to loss of minority-spin phase space.
Nonetheless, the rapid suppression of the T 2 contribution to the resistivity and the agreement between its
magnitude at higher temperatures with parameters expected for the manganites confirm the basic picture. In
the intermediate temperature regime, 0.2⭐T/T C ⭐0.5
here, the manganites appear to be normal metallic ferromagnets with the resistivity dominated by spin-wave
scattering. At lower temperatures, the increasingly half
metallic character of the material is manifested by a
temperature dependent cutoff of the spin-wave scattering process, leaving in its wake only residual resistivity
from the intrinsic doped-in disorder and indistinguishable phonon and two-magnon contributions. As these
heavily doped materials have significant disorder and
large residual resistivities, we recall that strongly disordered materials also exhibit T 2 regimes below half the
Debye temperature (Nagel, 1977). This result, however,
an extension of the Ziman theory of liquid metals, sets
an upper limit of ␣ T 2 / ␳ 0 ⯝0.03 before the resistivity
changes to a linear temperature dependence; our ratio is
unity at 100 K with no evidence for a linear regime. We
conclude that the quadratic temperature dependence is
not due to phonon scattering in a strongly disordered
An alternative explanation of the low-temperature
electrical transport is based on an argument by Alexandrov and Bratkovsky that polarons remain the predominant charge carriers even below T c (Alexandrov and
Bratkovsky, 1999), and that transport is dominated by
polaron tunneling. As a consequence, the lowtemperature resisitivity is predicted to behave as
␳ 共 T 兲 ⫽ ␳ 0 ⫹E ␻ s /sinh2 共 ប ␻ s /2k B T 兲 ,
where E is a constant and ␻ s is the softest optical mode.
This expression gives a T 2 contribution above T
⫽ប ␻ s /k B , which as been determined to be 86 K from
fitting to data on polycrystalline La0.8Ca0.2MnO3 (Zhao
et al., 2000). However, as we have seen in Fig. 24, the
lowest optical mode has an equivalent temperature of
250 K, considerably higher than the polaron picture
would indicate.
In a recent theoretical effort, Furukawa (2000) has
discussed the problem of quasiparticle lifetimes due to
magnon scattering processes in half metals, taking into
account the effects of spin fluctuations. He argues that
there is no minority-spin Fermi surface in a half metal at
zero temperature but that spin fluctuations induce a minority band at finite temperatures. At low temperatures,
then, the one-magnon scattering self-energy depends on
spin fluctuations and increases as T departs from zero.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
This nonrigid band picture allows unconventional onemagnon scattering processes to arise as the minority
band is created and occupied, provided the incoherent
limit ␦ m⫽ 关 „M(0)⫺M(T)…/M(0) 兴 Ⰶ1 holds. In this
limit the resistivity is expected to be proportional to the
cube of the temperature divided by the spin-wave stiffness constant ␳ incoherent ⬀(T/D) 3 , where D is defined
via ␻ q ⫽Dq 2 . As the temperature increases, ␦ m cannot
be considered small and conventional magnon scattering
is recovered (coherent limit), where the usual T 2 dependence in the resistivity is expected. The crossover
temperature is estimated by T * ⯝ 关 2M(0)/0.06W 兴 2 D 3
where W is the bandwidth, M(0) the saturation magnetization, and D the spin-wave stiffness. Reasonable parameters for manganites result in T * ⬇50 K, in good
agreement with the temperature where the T 2 regime
appears in La0.67(Ca, Pb)0.33MnO3 (Jaime et al., 1998),
La1⫺x Srx MnO3 (0.2⬍x⬍0.4) (Furukawa et al., 2000),
and bandwidth controlled La0.67(Ca, Sr)0.33MnO3
(Broussard et al., 1999). Similar results have also been
observed in a classical half metal, CrO2 (Watts et al.,
The Hall effect in the ferromagnetic phase of manganites is particularly puzzling, and a relatively large effort
has been made to try to understand it. In ferromagnetic
metallic systems, the embedded magnetic moments
cause asymmetric scattering of current-carrying electrons, producing a voltage that superimposes on the
usual Hall voltage, the so-called anomalous Hall voltage.
The total Hall resistance ␳ xy can be written as
␳ xy 共 B,T 兲 ⫽R H 共 T 兲 B⫹ ␮ 0 R S 共 T 兲 M 共 B,T 兲 ,
where R H (T) is the temperature-dependent Hall coefficient, B is the applied magnetic field (unit demagnetization factor assumed), R S (T) is the temperaturedependent anomalous Hall coefficient, and M(B,T) is
the magnetization. Measurements in films of composition La0.67Ca0.33MnO3 (Matl et al., 1998; Jakob et al.,
La0.67(Ca, Pb)0.33MnO3 (Chun et al., 1999) show an almost temperature-independent Hall coefficient R H and
strongly temperature-dependent anomalous Hall coefficient that peaks above the Curie temperature and is not
simply related to the ordinary resistivity. Typical results
in single-crystal samples are displayed in Figs. 27(a) and
(b). Figure 27(a) shows the measured Hall resistance in
magnetic fields up to 7 T, for several temperatures below
and above T C . At low temperatures (solid symbols) two
different regimes (low fields⬅anomalous, high fields
⬅normal) are evident. Figure 27(b) shows in its inset
how the two components are separated using the
demagnetizing-field-corrected magnetization measured
in the same samples. Also in the inset is shown the scaling of the anomalous hall coefficient with the sample
resistivity. The body of Fig. 27(b) displays the calculated
effective carrier concentration n eff(T)⫽1/eR H , which
show a monotonic increase as the temperature is reduced and is as large as 2.4 holes/Mn (almost five times
larger than nominal doping level). These large values
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 27. Hall data taken over a wide range of fields and temperatures: (a) Hall resistivity ␳ xy of a La2/3(Pb, Ca)1/3MnO3
single crystal as a function of field; (b) effective number of
holes per Mn atom as a function of temperature, calculated
from the data of (a), assuming the high-field slope to determine the ordinary coefficient, and n eff⫽e/R0 . The lower inset
shows the decomposition of ␳ xy into ordinary and anomalous
Hall effects below T C . The upper inset shows the linear relation between the anomalous Hall coefficient R S and the longitudinal resistivity ␳ xx . From Chun et al., 1999.
have been attributed to charge compensation effects. In
a two-band model, the Hall coefficient is given by R H
⫽(r h n h ␮ h2 ⫺r e n e ␮ 2e )/e(n h ␮ h ⫹n e ␮ e ) 2 and the experimental values in Fig. 27(b) can be reproduced using
M. B. Salamon and M. Jaime: Manganites: Structure and transport
band-structure calculations (Pickett and Singh, 1997b)
and a mobility ratio ␮e /␮h⫽1.6. The anomalous Hall
coefficient and its scaling with the sample magnetization
will be discussed in detail later, since it is closely related
to the physics of the transition between a hightemperature state where charge carriers are localized as
small polarons and the low-temperature ferromagnetic
state. Interestingly, layered manganites do not show
charge compensation effects and the carrier concentration deduced from Hall-effect measurements in single
crystals agrees very well with the values expected from
the nominal doping level. The anomalous Hall effect in
layered manganites seems to scale with the resistivity,
exhibiting a minimum at the same temperature where
the electrical conductivity is maximal (Chun, Salamon,
Jaime et al., 2000), but more work needs to be done extending the measurements above the Curie temperature
to be able to compare them to their 3D counterparts.
3. Thermal properties
a. Thermal conductivity of 3D materials
The thermal conductivity ␬ of ferromagnetic manganites, both in polycrystal (Hejtmanek et al., 1997; Chen
et al., 1997; Cohn et al., 1997) and single-crystal samples
(Cohn et al., 1997; Visser et al., 1997), is relatively small,
with room-temperature values between 1 and
3 W K⫺1 m⫺1 and a low-temperature peak that never exceeds 4 – 5 W K⫺1 m⫺1. These conductivities are 5–10
times smaller than those observed in the normal state of
high-temperature superconductor single crystals at similar temperatures (Peacor et al., 1991). In both systems
electrons contribute a small (20–40 %) fraction of the
total observed thermal conductivity ␬ tot in the hightemperature phase. In high-temperature superconductors the mean free path of unpaired electrons increases
dramatically below the superconducting transition, causing ␬ e to dominate ␬ tot somewhat below the transition
before dropping at low temperatures. In manganites the
transition into the metallic ferromagnetic state shows a
similar effect, i.e., the thermal conductivity improves as
the electrons (localized in the paramagnetic state) gain
mobility in the ferromagnetic phase but ␬ e remains a
small fraction of ␬ tot . Figure 28(a) displays the thermal
conductivity in La1⫺x Cax MnO3 polycrystals for different
␬ values, Fig. 28(b) shows the thermal conductivity vs
temperature in semilog scale for a number of polycrystal
and single-crystal samples. In the low-temperature region (the high-temperature paramagnetic region will be
discussed later) we notice an appreciable enhancement
of the thermal conductivity in ferromagnetic compounds. The Wiedemanz-Franz law can be used to estimate the electronic contribution to the thermal conductivity ␬ e in the metallic state. Such an estimate in Cohn’s
samples results in an electric contribution no larger than
20% of the total thermal conductivity. A similar estimate
for single crystals and films of different Curie temperatures is displayed in Fig. 29(a), where it can be seen that
␬ e (T) can only qualitatively explain the lowtemperature increase in ␬. In order to make a quantitaRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 28. Comparison of thermal conductivity for various
samples: (a) thermal conductivity vs temperature for
La1⫺x Cax MnO3 polycrystals. The inset is a magnified view of
the data for x⫽0 and 0.15, showing the anomaly observed near
T C ⫽170 K for x⫽0.15. The solid line represents the data for
x⫽0 shifted upward by 0.02 W/mK (Cohn et al., 1997). (b)
Thermal conductivity ␬ for four different manganite perovskite samples. The arrows denote the ferromagnetic transition
temperature. The three samples possessing a sharp change in
␬ (T) at T C also undergo a metal-insulator transition. The
dashed lines represent an exponential temperature dependence ␬ ⫽ ␬ 0 exp(T/350), with ␬ 0 values described in the text.
The inset shows the electrical conductivity ␴ versus ␬, at 350 K
for the four samples. From Visser et al., 1997.
tive analysis, we have taken data from Visser’s report
and plotted them together with the result of a
Wiedeman-Franz estimate and an extrapolation of the
high-temperature phonon contribution in Fig. 29(b).
Here we can see that ␬ e (T) can only account for 40% of
the low-temperature value by itself and about 80% of
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 29. Separation of thermal conductivity into electron and phonon contributions: (a) Electronic thermal conductivity calculated
from electrical resistivity in a La0.67(Ca,Pb)0.33MnO3 single crystal using the Wiedemann-Franz law. (b) Total thermal conductivity
␬ exp for La0.2Nd0.4Pb0.4MnO3 single crystal (from Visser et al., 1997), high-temperature phonon contribution ␬ ph and electronic
contribution ␬ e calculated using the electrical resistivity and Wiedemann-Franz law. (c) Temperature dependence of the total
thermal conductivity ␭ total ⫽␭ electron ⫹␭ phonon for a ferromagnetic (T C ⫽205 K) Pr0.6Ca0.2Sr0.14MnO3 single crystal. For the separation of phonon and electron parts of thermal conductivity the electronic contribution was estimated using the experimental
resistivity data, corrected for sample porosity and assuming the validity of Wiedemann-Franz law. (d) Similar data for a layered
compound. Note different scales for in-plane and out-of-plane curves. From Hejtmanek et al., 1999.
the total when added to a simple extrapolation of the
high-temperature lattice contribution ␬ ph . In Fig. 29(c)
we see data for a single crystal of composition
Pr0.65Ca0.21Sr0.14MnO3 (Hejtmanek et al., 1999) from
which it is clear that an extra contribution is missing
from the picture. Two conclusions can be derived from
this behavior. In the first place, the lattice thermal conductivity in the ferromagnetic manganites is lower than
in the cuprates and is not limited by electron-phonon
scattering. This conclusion finds support in the absence
of a linear term in the low-temperature electrical conductivity. Consequently we believe that the observed ␬ ph
is small due to the presence of large cationic disorder
and to the scattering of phonons against Jahn-Teller
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
modes (an estimate by Cohn et al. found that the
phononic mean free path is comparable to the Mn-Mn
distance ⬃4 Å). Second, because of the very nature of
small magnetoelastic polarons in the paramagnetic
phase, as the temperature is reduced and the charge carriers delocalize, the effective electron-phonon scattering
rate decreases and both thermal conductivities (␬ e and
␬ ph ) increase in the metallic state.
b. Thermal conductivity of layered materials
The thermal conductivity of single crystal samples of
the layered manganite La1.2Sr1.8Mn2O7 is anisotropic, as
expected from the structure and the behavior of other
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 30. The Seebeck coefficient S vs temperature for a mixed
Pb-Ca-doped sample. It is positive and metal-like at low temperature, has an anomalous kink near 30 K, and develops a
positive field dependence above 40 K. The dashed line is a
linear fit in the low-temperature regime. From Jaime et al.,
transport properties (Matsukawa et al., 2000). The component along the ab crystallographic direction increases
in the ferromagnetic state as in the 3D compounds,
while the component along the c axis shows no change.
Another contribution to ␬ tot in manganites should come
from spin waves ( ␬ m ), however, an estimate using the
specific-heat anomaly ⌬C at T C results in a very small
␬ m ⯝0.1 W K⫺1 m⫺1
La0.7Ca0.3MnO3 (Cohn et al., 1997). Detailed measurements of the very low-temperature thermal conductivity
in good single crystals should tell us more about the
relative contribution from spin waves, and could help to
better understand the transport properties of manganites. A discussion of the high-temperature thermal conductivity in manganites is presented in Sec. IV.
c. Thermoelectric properties and heat capacity
The Seebeck coefficient S(T) provides additional information on the nature of transport at low temperatures. Figure 30 shows S(T), measured on the same
sample as in Fig. 26 at H⫽0 and 80 kOe. At the lowest
temperatures, S(T) is positive, linear in temperature,
and extrapolates to zero as T→0. The field dependence
is small and negative. The large slope suggests, from the
Mott formula, that the resistivity is a strong function of
energy at E F . There is a sharp deviation from linear
behavior in the temperature range in which the T 2 dependence of the resistivity becomes dominant and the
field dependence changes sign and becomes larger. We
note that the peak in the low-temperature thermopower
that is regularly seen in thin-film samples is absent here,
and is therefore not intrinsic to these materials. If we
take the scattering to be independent of energy, which is
the case below 20 K, the Seebeck coefficient can be ex2
pressed as S(T)⫽( ␲ 2 /2e)(k B
T/E F ) (Ashcroft and Mermin, 1976). Using the simplistic approximation of a
parabolic band E F ⫽ប 2 k F2 /2m * , and spherical Fermi surRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 31. Composition dependence of the heat capacity: (a)
From top to bottom, low-temperature specific heat, plotted as
C/T vs T 2 , for LaMnO3⫹ ␦ : 䊊, with ␦ ⫽0.26; 䉮, ␦ ⫽0.15; 䉭,
␦ ⫽0.11; for La1⫺x Cax MnO3 䊐, x⫽0.11 〫, x⫽0.33 Ghivelder
et al., 1999. (b) Specific heat C for La1⫺x Srx MnO3 , x⫽0.15
under magnetic fields. The inset shows reduction of C upon
application of 9 T for x⫽0.15. From Okuda et al., 1998.
face k F3 ⫽3n ␲ 2 , the effective mass turns out to be
m * /m⬇3.7, comparable to the value obtained from
specific-heat measurements. The sharp deviation from
linear behavior in the temperature range 20–40 K correlates with the onset of electron-magnon scattering
which, being a spin-flip process, must involve the minority spin band, and which therefore has a different dependence on energy near E F . Comparable behavior has
been observed in CrO2 films (Watts, 2000).
The specific heat in the ferromagnetic phase of manganites has received a great deal of attention recently.
Measurements at low temperatures in La1⫺x Cax MnO3
have been done in polycrystalline samples (Coey et al.,
1995; Hamilton et al., 1996; Ghivelder et al., 1998). Results for x⫽0.1, 0.33, and 0.62 are displayed in Fig.
31(a) between 4 and 10 K. Both experimental groups
have attempted to fit the data using three-parameter fits
of the form
C⫽C e ⫹C l ⫹C m ,
where C e is the electronic contribution ␥ T, C l is the
lattice contribution ␤ T 3 ⫹ ␣ T 5 , and C m is the magnetic
M. B. Salamon and M. Jaime: Manganites: Structure and transport
contribution, ␦ T 3/2 in the case of a ferromagnet, ␦ 2 T 2 in
the case of a standard Néel antiferromagnet. A fit to the
experimental results for metallic La0.67Ca0.33MnO3 gives
␥ ⫽4.7 mJ/mol K2, ␤ ⫽0.12 mJ/mol K4, and ␦ ⫽0 indicating the absence of a spin-wave contribution. The electronic density of states at the Fermi energy determined
from the measured ␥ is N(E F )⫽3 ␥ /( ␲ k B ) 2
⫽2.0 eV⫺1 Mn⫺1. The lattice contribution indicates a
Debye temperature ␪ D ⫽430 K. Neutron-scattering measurements (Lynn et al., 1996) provide a direct measurement of the spin-wave stiffness constant D
⫽170 meV Å2 , which would yield a specific-heat magnetic term ␦ ⫽0.62 mJ/mol K5/2, equivalent to 20% of the
experimental value at 5 K and above the scatter of the
data. The absence of a spin-wave component in the lowtemperature specific heat is not understood at present.
On the other hand, measurements in the system
La0.7Sr0.3MnO3 do show a spin-wave component, see
Fig. 31(b) (Coey et al., 1995; Woodfield et al., 1997). The
stiffness constant deduced from it is D⫽130 meV Å2 in
rough agreement with the neutron-scattering results D
⫽188 meV Å2 (Martin et al., 1996) and D⫽154 meV Å2
(Smolyaninova et al., 1997), with no indication of a spinwave energy gap. The electronic component in this material is ␥ ⫽3 – 7 mJ/mol K2 implying a density of states
at the Fermi energy N(E F )⫽1.3– 2.7 eV⫺1 Mn⫺1, close
to the value found in La0.33Ca0.67MnO3 and roughly a
factor 2–3 larger than the band-structure estimate (Pickett and Singh, 1996) N(E F )⫽0.8 eV⫺1 Mn⫺1, implying
some mass renormalization effect is present. The study
of the doping dependence of the Sommerfeld coefficient
␥ in La1⫺x Srx MnO3 (Okuda et al., 1998) reveals a behavior not typical for a filling-controlled metal-insulator
transition. Results for La1⫺x Srx TiO3 and La1⫺x Srx MnO3
are displayed in Fig. 32. The critical mass enhancement
near the metal-insulator phase boundary (vertical hatching), indicating a canonical Mott transition, in
La1⫺x Srx TiO3 is absent in La1⫺x Srx MnO3.
A recent heat-capacity study of other compounds reveals more about the manganites behavior. Measurements in Nd0.67Sr0.33MnO3 (T C ⬇200 K) show no spinwave contribution as in the case of La0.33Ca0.67MnO3,
but contain a hyperfine contribution ⬃T ⫺2 , a broadened Schottky-like anomaly originating in the Nd spin
system and excess entropy attributed to Nd-Mn exchange interaction and responsible for a temperaturedependent ‘‘linear’’ term ␥ (T)T (Gordon et al., 1999).
Oxygen deficient samples of LaMnO3 have been measured by the group in Brazil (Ghivelder et al., 1999), in
samples with insulating, ferromagnetic, low-temperature
phases [Fig. 31(a)]. These compounds, as in the case of
La0.7Ca0.3MnO3, do not show a spin-wave contribution.
The most surprising feature in both Nd0.67Sr0.33MnO3
and LaMnO3⫺ ␦ is an unexpectedly large zerotemperature linear term, even in the case of insulating
compounds. Ghivelder et al. (1999) propose that the
spin-wave component and the large electronic component have their origin in ferromagnetic clusters, where
mass-enhancement mechanisms (magnetic polarons, latRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 32. Doping level (x) dependence of the electronic
specific-heat coefficient ␥ for La1⫺x Srx MnO3 compared with
that for La1⫺x Srx TiO3 (LaTiO3⫹x/2). The hatched vertical bar
for La1⫺x Srx TiO3 (LaTiO3⫹x/2) indicates an antiferromagnetic
metallic region sandwiched by AF insulating (x⬍0.05) and
PM metallic (x⬎0.08). The hatched region for La1⫺x Srx MnO3
is the metal-insulator transitional region. From Okuda et al.,
tice polarons, Jahn-Teller effect, or Coulomb interactions) are at play. It seems strange though that mass enhancement mechanisms have no effect in Ca or Sr compounds, nor in layered manganites as discussed below.
Heat-capacity experiments have also been performed
in layered manganite La1.3Sr1.7Mn2O7 (Okuda et al.,
1999). In this system spin-wave excitations contribute to
the specific heat and can be suppressed by a 9-T magnetic field, which results in a 12-K energy gap in the
spin-wave spectrum as shown in Fig. 33. The effect is
larger than in the case of 3D compounds, but not as
large as expected for an ideal 2D system. The authors
claim that a finite dispersion of the spin wave along the
c axis will explain the results. The zero-field linear term
is ␥ ⫽3⫾1 mJ/mol K2, comparable to values in the 3D
counterpart La0.7Sr0.3MnO3. The lack of mass enhancement effects in the linear term may imply the presence
of an anomalous carrier scattering process or dynamical
phase separation (Yunoki, Moreo, and Dagotto, 1998).
A relatively large low-temperature magnetoresistance
indicates that transport is somewhat limited by electronmagnon scattering, but spin-valve effects and fielddependent localization also play a role.
In conclusion, the low-temperature thermal properties
of manganites and layered manganites are still far from
being completely understood, especially in what concerns spin-wave contributions and mass enhancement
effects. More detailed quantitative studies are in order,
to test the spin-wave excitation spectrum in those compounds that show the contribution, in single-crystal
samples that reveal intrinsic as opposed to sampledependent effects. A summary of heat-capacity results is
shown in Table II.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 33. Heat capacity of layered manganites: (a) temperature
dependence of specific heat (C) as shown in a C/T vs T 2 plot
in 0 and 9 T for a layered (n⫽2) manganite; solid line the
extrapolated C/T value at 0 K, which expresses the ␥ value;
(b) reductions of C upon the application of a field of 9 T for
(LSMO327, x⫽0.35),
La0.6Sr0.4MnO3 (LSMOI13, x⫽0.4) along with the calculated
results (solid lines) for the simple-square 2D lattice and
simple-cubic 3D lattice with an appropriate spin-wave stiffness
constant (D⫽190 meV Å2 ). From Okuda, Kimura, and
Tokura, 1999.
tions become negligible and electric charge/lattice interplay is dominant (Jaime and Salamon, 1999).
As noted in Sec. II.B.2, the formation and transport
properties of small lattice polarons in strong electronphonon coupled systems, in which charge carriers are
susceptible to self-localization in energetically favorable
lattice distortions, were first discussed in disordered materials (Holstein, 1959) and later extended to crystals
(Mott and Davis, 1971). Emin (1973) and Hillery et al.
(1988) considered the nature of lattice polarons in magnetic semiconductors, where magnetic polarons are carriers, self-localized by lattice distortions but also dressed
with a magnetic cloud. A transition from large to small
polaron occurs as the ferromagnet disorders, successfully explaining the metal-insulator transition observed
experimentally in EuO. If the carrier, together with its
associated crystalline distortion, is comparable in size to
the cell parameter, the object is called a small, or Holstein, polaron. Because a number of sites in the crystal
lattice can be energetically equivalent, a band of localized states can form. These energy bands are extremely
narrow, and the carrier mobility associated with them is
predominant only at very low temperatures. At high
temperatures the dominant transport mechanism is thermally activated hopping, with an activated mobility ␮ p
⫽ 关 x(x⫺1)ea 2 /h 兴 (T 0 /T) s exp关⫺(WH⫺J3⫺2s)/kBT兴 with
a the hopping distance, J the transfer integral, x the
polaron concentration, and W H one-half of the polaron
formation energy. There are two physical limits for these
hopping processes, depending on the magnitude of the
optical phonon frequency. If lattice distortions are slow
compared to the charge carrier hopping frequencies, the
hopping is adiabatic, otherwise it is nonadiabatic. In the
adiabatic limit, s⫽1 and k B T 0 ⫽h ␻ 0 , where ␻ 0 is the
optical phonon frequency and in the nonadiabatic limit,
we have s⫽3/2 and k B T 0 ⫽(pJ 4 /4W H ) 1/3. The polaronic
transport in manganites is usually considered adiabatic,
in this case the conductivity is given by
A. Polaron effects—3D materials
In the early experimental study of manganites unexpected high-temperature transport properties were believed to be dominated by nonintrinsic effects like defects, crystalline disorder, grain boundaries, and
impurities. Years later, with the preparation of good
quality films by laser ablation on lattice-matched substrates and the growth of large single crystals, it became
evident that the observed behavior is intrinsic and due
to localization of charge carriers in small polarons
(Ohtaki et al., 1995; Jaime, Salamon, Pettit et al., 1996;
Jaime et al., 1997; Palstra et al., 1997; Chun et al., 1999).
The localization is a consequence of a large electronphonon interaction, enhanced by the Jahn-Teller activity
of Mn3⫹ in the manganites, and has an impact on the
electric and thermal transport properties as well as on
the lattice properties. We will see in what follows that
the high-temperature regime is where magnetic correlaRev. Mod. Phys., Vol. 73, No. 3, July 2001
⑀ 0 ⫹W H ⫺J
x 共 1⫺x 兲 e 2 T 0
exp ⫺
k BT
␴ 0T 0
exp ⫺
k BT
The temperature dependence observed in the hightemperature resistivity of manganites follows this adiabatic prediction very well, as seen in Fig. 34, from temperatures close to T C up to 1200 K (Jaime, Salamon,
Rubinstein et al., 1996; Worledge et al., 1996; De Teresa
et al., 1998). At high enough temperature magnetic correlations can be completely ignored, since charge/lattice
and charge/charge interactions dominate. In this regime
on-site Coulomb repulsion (Worledge et al., 1998) has
been observed in support of the small polaron picture.
Small grain polycrystalline samples, very thin and unannealed films, on the other hand, have been reported to
show variable-range-hopping-type localization (Coey
et al., 1995; Ziese and Srinitiwarawong, 1998) and nonadiabatic small polaron transport (Jakob, Westerburg
et al., 1998).
M. B. Salamon and M. Jaime: Manganites: Structure and transport
TABLE II. Results from fits to the low-temperature specific heat in manganite compounds. Units are mJ/mol K2 for ␥, K for ␪ D ,
meV Å2 for D sw , mJ/mol K3 for ␦ 2 , and J K/mol for A. The cases where a contribution from spin waves is observed, but its
magnitude not reported, are indicated by *. The cases where a spin wave contribution is not observed are indicated by -.
Order/T C
D sw
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Okuda et al., 1998
Woodfield et al., 1997
Woodfield et al., 1997
Woodfield et al., 1997
Coey et al., 1995
Hamilton et al., 1996
Hamilton et al., 1996
Coey et al., 1995
Ghivelder et al., 1998
Ghivelder et al., 1998
Ghivelder et al., 1998
Smolyaninova et al., 1997
Smolyaninova et al., 1997
Smolyaninova et al., 1997
Hamilton et al., 1996
Coey et al., 1995
Woodfield et al., 1997
Ghivelder et al., 1999
Ghivelder et al., 1999
Ghivelder et al., 1999
Gordon et al., 1999
Pr0.65Ca0.35MnO3 (H⫽8.5 T)
Okuda et al., 1999
⫹b .
e k BT
Contrary to the case of band semiconductors, where
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
The thermopower of small polaronic systems is similar
to that of band semiconductors, governed by thermal
activation of carriers across a small barrier and thus a
function of the inverse temperature:
the chemical potential determines the temperature dependence of both the conductivity and the thermoelectric properties and ⑀ 0 ⫽E ␴ ⫽E S , small polaronic conduction results in very different characteristic energies, i.e.,
E S ⫽ ⑀ 0 ⰆE ␴ because E ␴ includes W H , which implies
that for the charge carrier to be able to hop from one
site to another, a lattice distortion of energy 2W H needs
to be provided. In a band semiconductor the strongest
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 34. The resistivity of an n⫽⬁ manganite, following the
adiabatic, small-polaron hopping model at all stages of annealing. Each line is the cooling curve after being held at T h for 10
h. T h ⫽650 K for the top curve, and increases by 50 K for each
successive run, up to T h ⫽1200 K for the bottom curve. From
Worledge et al., 1996.
temperature dependence comes from the availability of
carriers in the conduction band; in a small polaron system the limiting factor is the number of available sites
that the charge carriers can hop to. For the manganites
high-temperature transport measurements reveal E S
⬇ few meVⰆE ␴ ⬇100– 200 meV, the signature of small
polarons (Fig. 35). The independent term k B b/e in the
thermopower is given by the configurational entropy of,
in the case of the manganites, placing a hole with spin
3/2 (S h ⫽3/2) moving in a spin-2 background (S b ⫽2),
namely, ⫺(k B /e)ln兵关2Sh⫹1兴/关2Sb⫹1兴其⫽⫺(kB /e)ln(4/5)
⫽⫺19 ␮ V/K plus the mixing entropy term that counts in
how many different ways x holes can be distributed between n Mn sites. The mixing term in the case of correlated 1D hoping with weak near-neighbor repulsion is
given by ln关x(1⫺x)/(1⫺2x)2兴 (Chaikin and Beni, 1976).
Alternative models were discussed by Heikes (1965),
giving in the correlated limit ln关(1⫹x)/(1⫺x)兴 and in the
uncorrelated limit (where double occupancy is allowed)
ln关x/(1⫺x)兴. None of these models can explain quantitatively the experimental results, and only the uncorrelated limit by Heikes predicts the right qualitative dependence on the carrier concentration x. There are
three proposed explanations for this behavior. The first
is a disproportionation model, where two Mn3⫹ atoms
generate Mn2⫹ and Mn4⫹ sites with the transference of
one electron. The disproportionation density is related
to oxygen nonstoichiometry (Hundley and Neumeier,
1997). Another possibility is that small polarons move
by hopping between divalent atoms in real space due to
the elastic stress introduced in the lattice by atomic size
mismatch (Jaime et al., 1997). This is a type of impurity
conduction where the number of available sites for hopping increases with the carrier concentration x and as a
consequence the mixing entropy remains unchanged. Finally, the Heikes uncorrelated limit also suggests that
multiple occupancy or collective behavior is possible for
small polarons.
One of the most distinctive properties of small poRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 35. The resistivity in the adiabatic limit and thermopower
vs 1000/temperature. Lines are fits in the high-temperature regime, and E ␳ ⫽ ⑀ 0 ⫹W H and E S ⫽ ⑀ 0 . As in Fig. 6, there is a
large difference in these two energy scales. From Jaime, Salamon, Pettit et al., 1996.
laronic transport is the magnitude and temperature dependence of its Hall mobility ␮ H (T). The Hall mobility
is not a power law of the temperature as in band semiconductors but thermally activated, with an activation
energy E ␮ calculated to be always less that for the drift
mobility E d . The simplest model predicts E ␮ ⬇E ␴ /3, experimentally observed in oxygen-deficient LiNbO3 (Nagels, 1980). As first pointed out by Friedman and Holstein (Friedman and Holstein, 1963) the Hall effect in
hopping conduction arises from interference effects of
nearest-neighbor hops along paths that define an
Aharonov-Bohm loop. When the loops involve an odd
number of sites sign anomalies arise in the Hall effect
and the Hall coefficient is activated with a characteristic
energy E H ⫽2E ␴ /3:
R H ⫽R H
冉 冊
2E ␴
3k B T
g H F 共 兩 J 兩 /k B T 兲
exp ⫺
⑀ 0⫹
4 兩 J 兩 ⫺E S
k BT
At sufficiently high temperatures, the contribution
from the anomalous Hall effect is sufficiently small to
allow a measurement of the polaronic contribution. Figure 36 (Jaime et al., 1997) shows both the sign anomaly
(negative Hall effect for hole transport) and activated
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 36. The magnitude of the Hall coefficient ⫺R H vs temperature showing the values obtained by 䉱, ramping the magnetic field up and 䉲, ramping; down in a thin-film sample; 䊏,
from a single-crystal sample with a higher transition temperature; dashed line, Arrhenius fit Inset: The natural logarithm of
the averaged Hall coefficient for the thin film. The solid line is
a linear fit giving 91 meV for the activation energy. From
Jaime et al., 1997.
behavior of the Hall coefficient expected from the
Friedman-Holstein picture in a film sample of
(La0.75Gd0.25)2/3Ca1/3MnO3. Indeed, the energy characterizing the exponential rise of the Hall coefficient, E H
⫽91⫾5 meV is about 2/3 the measured conductivity activation energy, E H /E ␴ ⫽0.64⫾0.03 in excellent agreement with theory.
In order to compare the prefactor R H
with the experimental results some assumptions need to be made. The
geometrical factor g d depends on the ratio of the probability P nnn of next-near-neighbor (nnn) hops to P nn ,
that of near-neighbor (nn) hops, through g d ⫽(1
⫹4P nnn /P nn ). If these probabilities are comparable,
i.e., if diagonal hopping is allowed in the Mn square sublattice, g d ⫽5, g H ⫽5/2, and the exponential factor in Eq.
(17) becomes exp关(ES⫺兩J兩)/3k B T 兴 ⯝1. In the regime 兩 J 兩
⭓k B T, the function F( 兩 J 兩 /k B T) is relatively constant
with a value ⬇0.2, leading to R H
⫻10⫺11 m3/C. This yields an estimated carrier density n
⫽3.3⫻1027 m⫺3, quite close to the nominal level of 5.6
⫻1027 m⫺3.
The main effect of Gd, trivalent as La, is to reduce the
Curie temperature by further reducing the Mn-O-Mn
bond angle (and hence reducing the bandwidth) without
changing the doping level. This trick made the Hall effect measurement possible to temperatures as high as
four times T C allowing the observation of the small polaron regime for the first time. Chun et al. have confirmed the picture with recent measurements in slightly
underdoped single crystals and show (Fig. 37) how the
small polaron picture breaks down as the Curie temperature is approached from above, a remarkable accomplishment given the small size of the samples and
the intrinsic experimental difficulties (Chun, Salamon,
Tomioka et al., 2000).
Note that diagonal hopping is not disallowed by symmetry considerations in manganites because the MnRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 37. The activated behavior of Hall coefficient R H (main
panel) and Hall mobility ␮ H (inset) above T C ⫽220 K for a
single crystal of La0.7Ca0.3MnO3. Above 300 K, the slopes are
in accord with small-polaron hopping theory. Also shown in
the main panel is the effective activation energy of the conductivity, E *
␴ . The Hall and resistivity data deviate from polaronic
behavior at the same temperature. From Chun, Salamon, Tomioka et al., 2000.
O-Mn bond angle is smaller than 180° (regardless of Gd
content). Diagonal hopping successfully explains the
sign anomaly in the Hall effect, which requires small
polarons traversing Hall-effect loops with odd number
of legs, predicts correctly the carrier concentration in the
sample and also reduces the value of the hopping attempt frequency required to fit the electrical conductivity prefactor (Jaime et al., 1997). Worledge et al. studied
the electrical resistivity of (Gd free) La1⫺x Cax MnO3
films in the full composition range 0⭐x⭐1 (Worledge
et al., 1998). They find that a model for small polarons
that hop adiabatically between nn sites and experience
Coulomb repulsion reproduces well the results but underestimates the total conductivity, and propose that
considering hopping to more than nearest neighbors
(nnn) could fix the problem. Interestingly, a theoretical
study based on ab initio density-functional calculations,
showing how cooperative Jahn-Teller coupling between
individual MnO6 centers in manganites leads to the simultaneous ordering of the octahedral distortion and
the electronic orbitals, has found that a nnn hoping integral V ␴⬘ ⫽⫺0.42 eV comparable to the nn hoping integral V ␴ ⫽⫺0.52 eV together with a small but finite third
neighbor hopping V ␴⬙ are necessary to fit the e g bands in
the linear muffin-tin orbital theory (Popovic and Satpathy, 2000).
The thermal transport of manganites in the paramagnetic state is particularly puzzling because, besides its
magnitude being very close to the theoretical minimum
value (Cohn, 2000) [see Fig. 28(a)], it does not show the
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 38. The inverse of the dc susceptibility for
La0.67Sr0.33MnO3: 䊊, experimental data. Solid and dotted lines
are fits to the constant-coupling and Curie-Weiss models, respectively. From Causa et al., 1998.
characteristic 1/T temperature dependence for heat carried by phonons with a mean free path limited by anharmonic decay (Berman, 1976). Instead ␬ (T) increases
with temperature. An exponential increase with equal
temperature parameters, found in samples with very different compositions, has been interpreted as due to dynamical lattice distortions, a consequence of
T-dependent Debye-Waller factors (Visser et al., 1997).
These results however have not been reproduced by
other groups (Hejtmanek et al., 1999; Cohn, 2000) and a
definitive explanation is still missing. One possibility is
that the very moderate increase in ␬ with temperature is
due to the increase of a relatively small electronic component added to a saturated and temperatureindependent phonon component. Such an increase of
the electronic component with temperature could be
due to the increase in thermally activated small polaron
mobility with temperature. Indeed, in Fig. 29(b) we can
see a small but distinguishable increase in the hightemperature electronic component ␬ e calculated using
the Wiedemann-Franz relation.
Electron-spin-resonance (ESR) studies in the paramagnetic phase reveal the nature and interaction properties of Mn3⫹-Mn4⫹ spin pairs. High-temperature ESR
experiments were performed by Causa et al. (1998) together with magnetic susceptibility experiments up to
1200 K in La0.67Ca0.33MnO3, La0.67Sr0.33MnO3,
Pr0.67Sr0.33MnO3, and La0.67Pb0.33MnO3 . ␹ dc (T) follows,
in all cases, a ferromagnetic Curie-Weiss temperature
dependence ␹ dc (T)⫽C/(T⫺⌰), at high enough temperatures (T⬎2⌰). For temperatures below 1.5⌰ the
curve for the inverse susceptibility vs T (Fig. 38) shows a
positive curvature suggesting spin clustering effects. In
the paramagnetic regime the ESR spectrum consists of a
single line, with g values that go from 1.992 for Ca compound to 2.00 for Pr-Sr compound, with a linewidth that
deviates from linearity and a tendency to saturation with
the temperature (Fig. 39). The computation for ␹ ESR
performed by Causa et al. demonstrates that both ␹ ESR
and ␹ dc have the same temperature dependence clearly
indicating that all the Mn ions contribute to the obRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 39. Peak-to-peak magnetic resonance linewidth ⌬H pp vs
temperature, showing a universal behavior for the perovskites
⬘ MnO3 with A ⬘ ⫽Ca, Sr, and Pb. Different symbols
La0.67A 0.33
refer to X-, L-, and Q-band data. The inset shows similar data
for the Pr-Sr analog. From Causa et al., 1998.
served ESR spectra and that the ESR linewidth should
be related to the relaxation mechanism of the coupled
magnetic system. The authors also claim that the temperature dependence of the ESR linewidth may be described by a universal curve, whose temperature scale is
associated with T C . The behavior above T C is determined solely by the temperature dependence of
␹ ESR (T) and the infinite temperature linewidth kept as
an adjustable parameter. No evidence is found of a spinphonon contribution to the experimental linewidth in
this regime. These results agree with previously discussed transport experiments that show evidence for decoupling of small polarons from spins above T⫽2T C .
Additional evidence for polaron formation is found
from structural and optical studies. Pair-distribution
analysis of neutron powder-diffraction data, the width of
EXAFS features, and Raman data all find evidence for
structural features consistent with polaron formation
well above T c . A surprising aspect of these results is
that such evidence persists well into the ferromagnetic
regime. We will return to the question of admixtures of
polaronic and metallic phases in Sec. V. Perhaps the
strongest evidence that polaron effects are important is
found in the isotope effect. Substitution of 16O by 18O
lowers the transition temperature by 21 K for
La0.8Ca0.2MnO3 (Zhao et al., 1996) and lesser amounts
for x⫽0.3 (Babushkina et al., 1999). The shift is attributed to the mass dependence of the polaron bandwidth,
and provides strong evidence that oxygen motions play a
central role in controlling the ferromagnetic transition.
B. Polaron effects—layered manganites
The paramagnetic state of layered manganites [small
n members of the Ruddlesden-Popper series
(La, A) n⫹1 Mnn O3n⫹1 ] has not received as much attention as the corresponding state in 3D compounds. There
are a few reasons for this. In the first place, single-crystal
samples are difficult to grow and multiple phases are
M. B. Salamon and M. Jaime: Manganites: Structure and transport
common (Berger et al., 2000); second, the thermal and
electric transport properties are strongly anisotropic and
results obtained in ceramic polycrystalline samples mix
up conductivity tensor components. Because of the
structural similarities between 3D and layered manganites (see Fig. 10) it is expected that charge-lattice correlations play an important role in both of them. However,
the tetragonal crystal field in n⫽2 compounds can split
the degenerate Mn e g orbitals, a condition that precludes the Jahn-Teller effect as the mechanism that
drives small polaron formation in the paramagnetic insulator phase of the n⫽⬁ materials. Nonetheless, measurements in single-crystal samples of La1.2Sr1.8Mn2O7
(Zhou et al., 1998) support the existence of charge localization effects due to small polaron formation. Anisotropic behavior is found in the electrical resistivity and
thermopower that indicate conduction by Holstein small
polarons in the c crystallographic direction (perpendicular to the La-Sr-Mn-O planes) and by ‘‘Zener-pair’’ polarons in the ab crystallographic direction (along La-SrMn-O planes) that condense into clusters as the
temperature is reduced. In a recent report, Liu et al.
show that electrical transport in polycrystalline samples
of La1.4Sr1.6Mn2O7⫹ ␦ also behave as predicted by the
small polaron Eqs. (14) and (15), with a thermal activation energy in the electrical conductivity close to 100
meV and a characteristic energy in the thermopower
roughly 15–20 times smaller (Liu et al., 2000). These
numbers are quite close to those found in the 3D compounds, implying similar small polaron formation energies W H are involved. The authors also report that N2
annealing of the samples reduces the sample resistivity
and thermopower and increases the Curie temperature
and magnetoresistance. Similar results are reported by
Jung in polycrystalline samples of La1.6Ca1.4Mn2O7
where E ␴ ⯝100 meV is substantially larger than E S
⫽13 meV (Jung, 2000). A common feature in many layered manganites, the magnetic transition and a maximum in the thermoelectric power occur well above the
temperature at which the electrical resistivity peaks
(Hur et al., 1998). This has been interpreted by several
authors as indicative of small polaron cluster formation.
Thermal-conductivity measurements were carried out
in La1.2Sr1.8Mn2O7 crystals (Matsukawa et al., 2000). The
thermal conductivity is anisotropic, being between three
and five times larger in the ab direction. The thermal
conduction in the ab direction resembles that of the cubic samples [Fig. 29(d)] with a positive slope above T C
and a feature at the onset of the ferromagnetic transition. Along the c direction the thermal conductivity increases as the temperature decreases, in agreement with
what one would expect for most solids as the phonon
mean free path increases, with no feature at the Curie
A. Theoretical background
The transition between the ferromagnetic state described in Sec. III.C and the high-temperature polaronRev. Mod. Phys., Vol. 73, No. 3, July 2001
like behavior discussed in Sec. IV motivated the current
resurgence of interest in these materials. The extreme
sensitivity of the transport properties, the resistivity in
particular, to applied magnetic field is, of course, the
essence of colossal magnetoresistance. As discussed in
Sec. II.C, the dependence of T C on the charge-carrier
bandwidth in a double exchange system, and the dependence of the bandwidth on temperature, led Kubo and
Ohata (1972) to the conclusion that the nature of the
phase transition would differ from the behavior expected for a 3D Heisenberg ferromagnet. More recently,
Furukawa (1995a) used the ferromagnetic Kondo lattice
model Hamiltonian for infinite spin and dimension,
具 ij 典 , ␴
共 c i†␴ c j ␴ ⫹H.c.兲 ⫺J
兺i ␴ i •m i ,
to model the double exchange system and confirmed the
linear dependence of T c on the bandwidth.
Within the context of the Kubo-Ohata and Kondo
lattice-model approaches (Furukawa, 1994, 1995b) the
resistivity below the magnetic transition decreases as
␳ 共 M 兲 ⫽ ␳ 共 M⫽0 兲共 1⫺CM 2 兲 ,
and is therefore constant above T C . However, materials
that exhibit colossal magnetoresistance are characterized by exponentially increasing resistivity as the temperature is reduced toward T c .
The failure of earlier models to produce semiconductive behavior was remedied in a calculation by Millis
et al. (1995) that takes into account fluctuations in the
angle ␪ ij ⫽cos⫺1(Sជ i•Sជ j /S2), where i and j are neighboring
sites on the lattice. The resistivity involves the correlation function 具 ␪ 0 ␦ 1 ␪ R ␦ 2 典 , where ␦ 1 and ␦ 2 are nearest
neighbors of sites labeled 0 and R. Millis et al. point out
that this differs from the usual Fisher-Langer resistivity,
which involves only 具 ␪ 0 ␦ 1 典 correlations and that, because
static mean-field theory ignores fluctuations, it misses
the upturn in resistivity. The resistivity calculated in the
dynamic mean-field model is shown in Fig. 40 along with
data from Tokura et al. (1994). The calculated resistivity
is several orders of magnitude smaller than observed
and requires an unreasonably large field to produce significant magnetoresistance. The conclusion is that a major element is missing from the model, namely, the JahnTeller interaction that characterizes the parent
compound. In fact, each Mn4⫹ site, lacking an e g electron, does not gain energy from a Jahn-Teller distortion
of its local environment. It is likely, then, that hole motion is correlated with an ‘‘anti-Jahn-Teller’’ distortion,
by which we mean an empty e g site that is undistorted
relative to neighboring occupied e g orbitals and that
takes the form of the polarons discussed in Sec. IV.
The effect of Jahn-Teller coupling was first treated by
Röder, Zang, and Bishop (1996), who added it to Eq.
(18) and demonstrated the tendency of carriers to become self-trapped once the effective Jahn-Teller coupling is sufficiently large. A subsequent calculation by
Millis and co-workers (Millis, Shraiman, and Mueller,
1996; Millis, Mueller, and Shraiman, 1996) reproduced
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 40. Resistivity calculated from the double exchange
model without electron-phonon interaction: solid line, the resistivity in zero field; dashed line, the resistivity in a field of
0.15T C , corresponding to 200 T. The inset displays data from
Tokura et al. Both the clculated absolute change in resistivity
and the magnetoresistance are much smaller than experiment.
From Millis et al., 1995.
the qualitative behavior observed in manganites near
optimal doping. Figure 41 shows the magnetoresistance
in this model for values of the effective Jahn-Teller coupling constant just below (upper panel) and just above
(lower panel) the critical value for self-trapping. The effective coupling parameter ␭⫽g 2 /tk depends not only
the Jahn-Teller coupling strength g, but also inversely
on the bandwidth through t and the lattice stiffness via
k. As the dopant atom M changes from Ca to Pb to Sr
in La2/3M 1/3MnO3 , the bandwidth increases as the MnO-Mn bond angle tends toward 180°. Concomitantly, ␭
decreases such that Ca- and Pb-doped samples are
above the critical value and Sr-doped samples, below.
Substitution of Nd for La also affects the bond angle,
and makes the system more prone to polaron formation.
Typically, tabulated values of ionic sizes are used to estimate the deviation of the crystal structure from ideal
perovskite structure through the tolerance factor f. As
in the original Kubo-Ohata calculation and as shown in
Fig. 42, the transition temperature depends on ␭, which
depends on the effective hopping matrix element t
which, in turn, depends on the short-range spin order.
This bootstrap mechanism was treated by Millis, Mueller, and Shraiman (1996) by means of a dynamical meanfield method which yields sharp, but continuous
(second-order) transitions.
B. Nature of the phase transition
Despite the enormous body of literature in this field,
relatively little effort has focused on the nature of the
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 41. The effect of including Jahn-Teller effects, characterized by an effective coupling parameter ␭ in the double exchange model. The temperature dependence of the resistivity
is shown for different values of magnetic field h for (a) ␭
⫽0.7 and (b) ␭⫽1.12. The parameter h is related to the physical field through h⫽g ␮ B S c H phys /t. Using g⫽2, t⫽0.6 eV, and
S c ⫽3/2, we find that h⫽0.01 corresponds to H phys ⫽15 T.
From Millis, Mueller, and Shraiman, 1996.
phase transition beyond the obvious changes in resistivity. As the transition temperature is decreased, whether
by doping or ion-size substitution, self-trapping becomes
more evident through the activated behavior of the conductivity. Once T C is reduced to 200 K or so, the transition becomes hysteretic and clearly first order. This is
readily seen in the data of Fig. 43 (Hwang et al., 1995),
where the tolerance factor (and along with it, T C ) is
reduced by substitution of Pr for La. Hysteresis on cooling and heating becomes increasingly significant as the
transition temperature decreases (and the CMR effect
increases). Even above that temperature, where there is
no strong hysteresis, the transition differs markedly
from that of a conventional Heisenberg ferromagnet.
Clear evidence for this is seen in the heat-capacity data
on polycrystalline La0.7Ca0.3MnO3(LCMO) by Park,
Jeong, and Lee (1997). Rather than broadening with increasing field, the specific-heat peak shifts to higher temperature with little change in shape. This has been confirmed in single crystals of the same composition (T C
⫽216 K) by Lin et al. (2000) as seen in Fig. 44. The heat
capacity peak is very sharp, suggesting a nearly first-
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 42. The dependence of the Curie temperature T C on the
effective Jahn-Teller parameter ␭. Because it is inversely proportional to the bandwidth, ␭ decreases as magnetic order sets
in, causing the Curie temperature to increase with decreasing
temperature. Millis, Mueller, and Shraiman, 1996.
order transition. This contrasts with the heat-capacity
peak of La0.7Sr0.3MnO3 (T C ⫽359 K) which is much
more lambda like (Lin et al., 2000) and which shifts relatively little in applied fields up to 1 T. A small-angle
neutron study (Ibarra et al., 1998) of a nominal
La2/3Ca1/3MnO3 sample (T C ⬇250 K) also showed unusual characteristics associated with the transition.
While the quasielastic line shape is Lorentzian as expected, the inverse of its width, which measures the ferromagnetic correlation length, shows no power-law increase over the range 1.02T C ⭐T⭐1.15T C . Rather, the
intensity of the quasielastic peak increases as T C is approached then drops abruptly, following closely the anharmonic contribution to the volume thermal expansion
(Ibarra et al., 1998). Inelastic neutron scattering on a
comparable sample finds that the spin-wave stiffness coefficient D does not tend toward zero as for a conventional ferromagnet, but rather retains 50% of its low
temperature value at T C [see Fig. 23(c)]. In place of the
collapse of spin-wave groups toward zero energy, a
strong diffusive peak appears which grows in intensity as
T C is approached from below and persists to at least
1.1T C (Lynn et al., 1996). A subsequent neutronscattering study by Fernandez-Baca et al. (1998) found
very similar behavior in Nd0.7Sr0.3MnO3 (T C ⫽198 K)
but more conventional behavior in Pr0.63Sr0.37MnO3
(T C ⫽301 K).
Relatively little attention has been paid to the critical
point properties of CMR manganites. This is not surprising considering the evidence presented above that the
transition is percolative in nature. As the transition temperature is increased, either by doping or by changes in
tolerance factor, the tendency for polaron formation decreases and the resistive behavior is metallic over the
entire temperature range. Referring to Fig. 45, we see
that the innermost contour of the ferromagnetic metal
regime represents the most metallic, and possibly the
most ‘‘normal’’ ferromagnetic transition. Indeed, a scaling analysis of a single crystal of La0.7Sr0.3MnO3 (T C
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 43. Progression of the resistivity and magnetoresistance
with changes in trivalent ion: top panel; ln ␳(T) in 0 and 5 T for
a series of samples of La0.7⫺x Prx Ca0.3MnO3 (x⫽0, 0.175, 0.35,
0.525, 0.6, 0.7) and La0.7⫺y Yy Ca0.3MnO3 (y⫽0.35 and 0.5).
Bottom panel, magnetoresistance (cooling data) for x⫽0,
0.175, 0.35, 0.525, 0.6, 0.7 specified as ln(␳0T⫺␳5T)/␳5T , with the
maximum in ␳ 0T indicated by arrows. The interpretation is that
increasing the Pr content increases the effective Jahn-Teller
coupling constant by reducing the hopping matrix element.
From Hwang et al., 1995.
⫽354 K) by Ghosh et al. (1998) gives critical exponents
␤ ⫽0.37, ␥ ⫽1.22, and ␦ ⫽4.25, as shown by the data collapsing shown in Fig. 46. These values differ from 3D
Heisenberg values by amounts outside experimental uncertainty, and must be regarded as effective exponents
that reflect some residual tendency for mixed-phase behavior. As expected, samples with 20% Sr doping (T C
⬇310 K) exhibit quite different effective critical exponents that are far from Heisenberg values (Mohan et al.,
1998; Schwartz, 2000). The pyrochlore CMR compound
Tl2Mn2O7 , which has no indication of polaron formation, exhibits critical exponents that are very close to
theoretical values for the 3D Heisenberg model (Zhao
et al., 1999). These results support the idea that the critical temperature, reflecting the combined effects of JahnTeller coupling, bandwidth, and lattice stiffness, serves
as a measure of the degree to which polaron formation
drives the material toward a first-order phase transition.
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 44. Heat capacity of La0.7Ca0.3MnO3 plotted vs temperature in external fields of 0, 10, and 70 kOe. Unlike conventional ferromagnetic transitions, the peak moves to significantly higher temperatures with applied field. The peaks do
not obey the usual critical-point scaling behavior. From Lin
et al., 2000.
Further evidence for an unconventional phase transition in LCMO is found in muon relaxation data (Heffner et al., 1996). The muon precession frequency decreases with increasing temperature, but tends to zero at
a temperature above the T C deduced from the magnetization. In addition to the precessing component, which
relaxes at a rate comparable to the precession frequency,
a second contribution is found that relaxes at a rate several orders of magnitude slower. Initial studies of the
slow component showed a peak in the relaxation rate at
the magnetization value of T C , but with nonexponential
shape. Below T C the relaxation function is a stretched
exponential of form exp关⫺(⌳t)1/2兴 , slowly evolving into
a simple exponential by 1.2T C . The authors suggest a
glassy state in which fast-relaxing and slow-relaxing
components coexist for times on the order of ⬃1 ␮ s.
More recently, Heffner et al. (2000) have combined neutron spin-echo and muon relaxation data to identify two
distinct components to the relaxation rate in LCMO, in
place of a stretched exponential. One component is
identified with an extended cluster and exhibits fast spin
dynamics and critical slowing down, which translates
into slower muon relaxation, and an increasing volume
fraction as temperature is reduced. This component is
observed only at small momentum transfer (large length
scales) in the spin-echo data. The second component is
characterized by slowly fluctuating Mn spins (rapid
muon relaxation), and a decreasing volume fraction.
These results strongly suggest that the ferromagnetic
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 45. Phase diagram in the plane defined by the average
A-site covalent radius 具 r co v 典 and divalent ion concentration x,
showing the various antiferromagnetic insulating (AFI), ferromagnetic metal (FMM), ferromagnetic insulator (FMI), and
charge ordered insulating (COI) phases. Inside the region labeled FMM, contours of T C are shown. The dotted lines show
the effect of Sr, Ca, and Ba substitution. The bold arrow indicates that the effective Jahn-Teller coupling, ␭, increases as T C
decreases. Charge order is most stable near x⫽0.5 and x
⫽0.6– 0.67. From Ramirez, 1997.
transition proceeds by means of changing volume fractions of rapidly and slowly relaxing local magnetization.
C. Two-phase behavior
An early suggestion that two-phase separation might
govern the colossal magnetoresistive regime was made
by Gor’kov (1998). Monte Carlo simulations of the
double-exchange model with Jahn-Teller coupling also
suggested phase segregation of ferromagnetic metal
from antiferromagnetic insulator regimes. It is necessary
to make a distinction here between phase separation, by
which is meant coexistence of ferromagnetic/conducting
and paramagnetic/insulating regions with constant carrier density, and charge segregation as suggested by
Yunoki, Moreo, and Dagotto (1998). An experimental
test of these ideas was carried out by Jaime et al. (1999)
using an effective-medium approach. The resistivity data
were analyzed in terms of a temperature and magneticfield-dependent concentration c(H,T) of the metallic
phase. The power-law resistivity of the metallic regime
and the activated conductivity of the polaronic regimes
were assumed to persist into the mixed phase regime
near the metal-insulator transition. Combining these
with the measured resistivity, Jaime et al. extracted the
metallic concentration using data from a thin-film
sample of La2/3Ca1/3MnO3 using effective-medium expressions. Because the film thermal conductivity is
dominated by that of the substrate, the thermal conductivity can be eliminated from the transport equation to
give a very simple expression for the thermoelectric
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 46. Critical-scaling plots of La0.7Sr0.3MnO3 below and
above the Curie temperature (T C ⫽354.0 K), using ␤ and ␥ as
noted in the text. Inset shows the same plot on a log scale. ⑀ is
equal to 兩 T⫺T C 兩 /T C . From Ghosh et al., 1998.
S 共 H,T 兲 ⫽
关 ␳ S ⫺ ␳ met S pol
␳ pol ⫺ ␳ met pol met
⫹ ␳ exp共 H,T 兲共 S pol ⫺S met 兲兴 ,
where the subscripts refer to polaron, metallic, and experimental values. As with the resistivity, the thermoelectric powers are fit in the two regimes and assumed to
characterize the two phase region. The extracted values
of c(H,T) and the fits to the thermopower are shown in
Fig. 47.
A number of other experimental probes provide further evidence for the mixed nature of the transition in
the CMR regime. EXAFS studies of a series of
La1⫺x Cax MnO3 samples, with x⫽0.21, 0.25, and 0.30
were performed by Booth et al. (1998). The meansquare width of the Mn-O bond-length distribution decreases as the temperature is reduced, approaching at
low temperatures the values for x⫽1, for which there is
no Jahn-Teller distortion. Similar results were found
from neutron pair-distribution-function data for
La0.8Sr0.2MnO3 by Louca and Egami (1999). As seen in
Fig. 48, the number of short Mn-O bond lengths changes
from approximately four (indicating polaronic elongation of the oxygen octahedra) at high temperatures toward six (no average distortion) at low temperatures.
Similar results for La1⫺x Cax MnO3 have been reported
by Billinge et al. in which the two-phase coexistence reRev. Mod. Phys., Vol. 73, No. 3, July 2001
FIG. 47. Two-phase analysis of resistivity and thermopower:
(a) The concentration of metallic regions c(H,T) extracted
from the resistivity data using the effective-medium approximation and assuming elongated conducting regions with
length-to-width ratio a/b⫽50; (b) Seebeck coefficient data and
results of a computation using the measured resistivity
␳ exp(H,T) and an effective-medium approximation as described in the text. Dotted lines show the low- and hightemperature fits. From Jaime et al., 1999.
gion is mapped out via pair-distribution-function analysis (Billinge et al., 2000). Particularly convincing evidence of two-phase coexistence is found in the Raman
data of Yoon (1998). The spectrum for a sample of
Pr0.7Pb0.21Ca0.09MnO3 exhibits, above its Curie temperature T C ⫽145 K, two features in addition to phonons: a
broad peak centered near 1100 cm⫺1 and a lowfrequency diffusive response. The peak is attributed to
the photoionization energy of small polarons and the
diffusive signal, to their hopping motion. As the temperature is reduced, the diffusive signal evolves into a
flat continuum response characteristic of metallic behavior. Figure 49 shows the temperature dependence of the
integrated Raman intensities for the diffusive (polaronic) and continuum (electronic) contributions. Their
coexistence in the vicinity of T C is readily apparent.
Quite recently, diffuse neutron-scattering peaks have
been detected (Dai, Fernandez-Baca et al., 2000) that indicate polaron-polaron correlations exist over a wide
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 48. Bond-length distribution: Top, the number N Mn-O of
short Mn-O bonds as a function of temperature for
La0.8Sr0.2MnO3, as determined by neutron pair-distributionfunction analysis. The presence of four short bonds at high
temperatures is indicative of the existence of Jahn-Teller polarons. The bond lengths tend to equalize with decreasing temperature, but some residual distortion of the oxygen octahedra
remains at low temperatures. Bottom, the height of the pairdistribution function peak at 2.75 Å which includes the O-O
pair. Its relation to temperature is similar to that of the N Mn-O
value. From Louca and Egami, 1999.
temperature range above T C , but collapse below. The
position of the peak suggests short-range charge ordering with a correlation length of ⬃1.4 nm at room temperature, increasing to ⬃2.8 nm at the transition. There
is no sign of long-range charge ordering. Further evidence for mixed phases comes from a Mossbauer study
of LCMO doped with 57Co (Chechersky et al., 2000).
Coexisting ferromagnetic and paramagnetic sites are
found in the vicinity of the phase transition, as might be
expected from the conducting/insulating mixture postulated by Jaime et al. (1999).
Additional evidence for inhomogeneous conductivity
in the vicinity of the phase transition is found on the
noise characteristics of film and single-crystal samples.
Early work (Hardner et al., 1997) on a partially annealed
La2/3Ca1/3MnO3 film (T C ⬇100 K) revealed presence of
significant non-Gaussian noise in the form of discrete
resistance switching that is largely absent well below the
transition, but predominant at T C . Anomalously large
1/f noise is also found in LCMO films (Alers et al., 1996;
Raquet et al., 2000), and it has been found that the magnitude in epitaxial films is sensitive to oxygen content
(and therefore T C ) (Rajeswari et al., 1998). Samples that
are in the first-order transition regime show very large
1/f noise which is taken as evidence for a percolation
transition (Podzorov et al., 2000). Recent noise measurements on single crystals also finds a large 1/f contribution in LCMO with evidence for two-state switching in a
narrow temperature range near T C ⫽218 K. From the
field and temperature characteristics of the switchers,
the authors conclude that regions as large as 104 unit
cells are fluctuating between high and low resistance
states, providing strong evidence for inhomogeneous
conducting paths as expected for a percolationlike transition (Merithew et al., 2000).
When the transition is fully first-order in character,
the phase separated regions are mesoscopic (micrometer
range) and have been observed by electron microscopy
(Uehara et al., 1999). Static images of conducting and
insulating regions have also been captured using
scanning-tunneling microscopy (Tath et al., 1999). In this
study, coexisting insulating and conducting regions on
the 100-nm scale were imaged, and the growth of metallic regions was monitored as a magnetic field was applied near the transition temperature. Such quasistatic
objects may be related to the slow-switchers observed in
noise studies, but are larger in scale that those detected
by neutron scattering. A recent calculation has indicated
that these giant clusters have equal electronic density
and result from disorder in the exchange and hopping
amplitudes in the vicinity of a first-order transition
(Moreo et al., 2000).
D. Phase-separation models
FIG. 49. Raman-scattering data showing the temperature dependence of the integrated scattering strength ratio for 䊉, the
diffusive response, I diff (T)/I diff (T⫽350 K) indicative of polarons, and 䊏, the flat continuum response, I cont (T)/I cont
(T⫽90 K) indicative of band electrons, for a T C ⫽145 K
sample. From Yoon et al., 1998.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
The effective-medium approach suggests that metallic
and insulating regions coexist as interpenetrating clusters, also suggesting a percolation picture of the
insulator-metal transition. One such percolative model
was proposed by Bastiaansen and Knops (1998) based
on a random resistor network. A Monte Carlo simulation of a 2D Ising model formed the basis of the calculation, with unit resistors connecting aligned nearestand next-nearest-neighbor sites and infinite resistance
linking unaligned sites. The resistance of the network is
qualitatively similar to experiment. For a system of
Heisenberg spins, of course, resistance of each link can
vary continuously between 1 (linked spins parallel) and
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 51. Inverse magnetic susceptibility H/m vs temperature
near the M-I transition for H⫽24 kOe. Dashed line, the zerofield extrapolated behavior. The appearance of free carriers
induces an increase in the effective T C , leading to a kink in
the susceptibility. Inset: inverse susceptibility of a single-crystal
sample of La0.7Ca0.3MnO3, measured in a field of 1 kG. From
Jaime et al., 1999.
Teller coupling constant ␭ eff is smaller than the critical
value ␭ c . Because ␭ eff is, in turn, inversely proportional
to the bandwidth, the double exchange mechanism
causes ␭ eff to decrease as magnetic order sets in. In the
CMR regime, where ␭ eff is presumed to be near ␭ c , it
can be expanded as
FIG. 50. Results of a coupled-order parameter model: (a) the
conducting concentration c(H,T) calculated in the mean-field
model with ␣ ⫽0.02 and ␥ ⫽0.3; (b) the magnetization calculated with the same parameters. The dotted line shows the
noninteractive case for comparison. From Jaime and Salamon,
infinity (linked spins antiparallel), greatly reducing the
size of the resistance peak just at T C . Quite similar results have been reported recently by Mayr et al. (2001).
A more realistic model was proposed by Lyukyutov and
Pokrovsky (1999) which is based on Varma’s theory
(Varma, 1996) of magnetic polaron formation, modified
to include Jahn-Teller effects. Magnetic polarons, which
coexist with small lattice polarons, are assumed to be
large, basically comprising magnetically correlated regions. As the temperature is lowered, the magnetic polaron density increases until the magnetic polarons overlap, which defines the percolation point. The authors
argue that long-range Coulomb effects render implausible suggestions that macroscopic charge separation underlies the CMR effect (Yunoki et al., 1998; Moreo et al.,
1999). Similar ideas have been discussed by Gor’kov
(Gor’kov, 1998; Gor’kov and Kresin, 1998).
A simple mean-field model has been proposed by
Jaime et al. (1999) that gives a qualitative description of
the conducting fraction c(H,T). A simplification of the
dynamic mean-field theory of Millis, Mueller, and
Shraiman (1996), it recognizes that the fully metallic
state c(0,T→0) is achieved only if the effective JahnRev. Mod. Phys., Vol. 73, No. 3, July 2001
␭ eff⯝␭ c ⫹ ␣ ⫺ ␥ m 2 ,
where ␣ is a small positive constant that assures that
polarons will form once the magnetization m
⫽M(H,T)/M sat becomes smaller than ␣ / ␥ . The conducting fraction is then taken to be an order parameter
c⫽tanh关共 1⫺ ␣ ⫹ ␥ m 2 兲 c 兴 .
The coupling constant ␥ causes the Curie temperature to
increase when c is finite, so that the magnetization in
zero field rises more rapidly for c⭓0. Figure 50 shows
the conducting fraction and magnetization at several
magnetic fields for ␣ ⫽0.02 and ␥ ⫽0.4, which compare
well with c(H,T) seen in Fig. 47(a). There is a kink in
the magnetization at the point at which the mixing factor
begins to increase; such kinks are frequently observed
experimentally as seen in Fig. 51 (Jaime et al., 1999).
This model cannot, of course, capture the essential contributions of magnetic/conducting fluctuations.
E. Hall effect in the transition region
Even in the regime where the charge carriers are
nominally ‘‘metallic,’’ estimates based on free-electron
ideas (electron density n⬇1027 m⫺3, effective mass
m⬇3m e ) suggest that the mean free path approaches
interatomic distances once the resistivity exceeds
1 m⍀ cm. Therefore even the metallic portions of any
percolation network must be considered to be localized
M. B. Salamon and M. Jaime: Manganites: Structure and transport
pend only on the magnetization, independent of the details of the hopping process, and is given by
␳ xy ⫽ ␳ xy
m 共 1⫺m 2 兲 2
1⫹m 2
Figure 52 shows that this is obeyed by the data of Fig.
that are in reasonable agree27(a) with values of ␳ xy
ment with estimates using atomic spin-orbit coupling
constants. The boundary between this anomalous contribution and the ordinary (Holstein) contribution was
found recently by Chun, Salamon, Jaime et al. (2000) as
a minimum in the Hall mobility.
A. 3D manganites
FIG. 52. Plot of the Hall resistivity ␳ xy vs reduced magnetization using the data shown in Fig. 27 showing scaling behavior:
solid line, a fit to Eq. (23); dashed line, the numerator of Eq.
(23) only. There are no fitting parameters except the amplitude. From Chun, Salamon, Lyanda-Geller et al., 2000.
(Gor’kov, 1998; Lyanda-Geller et al., 2000). The degree
to which the localization length shrinks to permit selftrapping by Jahn-Teller processes determines the
strength of the CMR effect. However, once the resistivity exceeds this Mott-Ioffe-Regel limit, all transport processes must be considered within a hopping framework.
This point is made forcefully by the strongly anomalous
behavior of the Hall effect which is very similar for a
variety of manganites despite major differences in their
resistive properties. Figure 27(a) shows the Hall resistivity of the single-crystal sample whose longitudinal resistivity was shown in Fig. 1. Despite the very large differences in ␳ xx among various samples, the ␳ xy data are
remarkably similar, and deviate strongly from the behavior of conventional ferromagnets in the critical region.
In conventional magnets, Eq. (13) permits the separation of ordinary and extraordinary components through
the critical point, here the Hall resistivity becomes nonmonotonic and strongly curved. Only at high temperatures, where the polaronic effects dominate, and at very
low temperatures, can an ordinary Hall coefficient be
defined, as seen in Figs. 27(a) and (b). The first suggestion that a mechanism beside skew scattering and sidejump processes was needed was made by Ye et al.
(1999). Formulated for metallic conduction, this model
predicted a maximum in ␳ xy above the Curie temperature, which is contrary to what is observed. Recognizing
that the entire transition region is in the nonmetallic,
hopping regime, Chun, Salamon, Lyanda-Geller et al.
(2000) extended the classic Hall calculation of Holstein
(1959) for the Hall constant of hopping charge carriers
to take account of the Berry-like phase that is accumulated by charge carriers that are required by strong
Hund’s-rule coupling to follow the local spin texture.
The model predicts that the Hall resistivity should deRev. Mod. Phys., Vol. 73, No. 3, July 2001
Away from the ferromagnetic regime, and even overlapping it somewhat, is a variety of charge and orbitally
ordered phases. The experimental situation has been reviewed in some detail recently by Rao and Raveau (Rao
et al., 2000). It might be argued that charge ordering at
doping levels that are rational fractions represent an
order-disorder transition of Jahn-Teller polarons. This
interpretation was favored by Cheong and Chen (1998)
who found evidence for charge ordering in
La1/2Ca1/2MnO3 and La1/3Ca2/3MnO3 . Indeed, looking at
Fig. 12 we would expect that approach to describe
Sm1/2Ca1/2MnO3 , Nd1/2Ca1/2MnO3 , and Pr1/2Ca1/2MnO3
and, for the last of these, direct evidence for it from
high-resolution electron microscopy has been reported
(Li et al., 1999). However, this picture cannot hold for
Nd1/2Sr1/2MnO3 or Pr1/2Sr1/2MnO3 , where the chargeordered phase arises within the more metallic, ferromagnetic regime. Figure 53 shows the contrast between
Sm1/2Ca1/2MnO3 and Nd1/2Sr1/2MnO3 (Tomioka et al.,
1997). While the former shows a weak peak in magnetization and an insulator/insulator transition at the charge
ordering temperature, the latter clearly becomes ferromagnetic and metallic near 250 K before undergoing a
first-order metal/insulator transition to an antiferromagnetic phase near 160 K. That Nd1/2Sr1/2MnO3 and
Pr1/2Sr1/2MnO3 both become antiferromagnetic at the
charge ordering transition is shown by the neutrondiffraction data (Kawano et al., 1997) in Fig. 54. That
figure also demonstrates the narrowness of the antiferromagnetic CE phase (see Fig. 11 for its structure)
shown in Fig. 17. This has been explored in some detail
by Kajimoto et al. (1999). At a Sr content of x⫽0.55 the
ferromagnetic phase does not intercede between the
paramagnetic state and the insulating, A-type antiferromagnetic phase. As was noted previously, the tendency
toward stabilization of Jahn-Teller polarons is strongly
correlated with the ionic size of the divalent dopant in
the A site. This has been attributed to the intrinsic bandwidth to be expected in the fully metallic state, which in
turn depends on the Mn-O-Mn bond angle. At x⫽1/2, as
well, the primacy of the tendency toward charge ordering and an insulating state is closely tied to the tolerance
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 53. Charge ordering transitions in Sm1/2Ca1/2MnO3 and
Nd1/2Sr1/2MnO3. NSMO exhibits a ferromagnetic, metallic
phase before undergoing a
charge ordering transition. In
contrast, SCMO makes a transition directly to the insulating,
charge ordered state. From Tomioka et al., 1997.
FIG. 54. Neutron-scattering results on the ferromagnetic-toantiferromagnetic transitions in x⫽1/2 manganites showing
A-type (Pr1/2Sr1/2MnO3) and CE-type (Nd1/2Sr1/2MnO3) order.
The latter result on NSMO supports the data shown in Fig. 53.
Off x⫽1/2 the ferromagnetic phase is absent and A-type order
occurs. From Kawano et al., 1997.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
factor. Figure 55 summarizes the sequence of transitions
for various combinations of A-site atoms at x⫽1/2.
A dramatic feature of the charge ordered state is that
it can be ‘‘melted’’ by a magnetic field, as first reported
by Kuwahara et al. (1995). The transition, which is hysteretic in temperature at zero field, becomes strongly
hysteretic in applied fields, as may be seen in Fig. 12.
The melting transition is clearly observed in the resistivity, as is shown in Fig. 56 for Pr1/2Ca1/2MnO3 . The
changes are even more dramatic (Tokura et al., 1996) for
Nd1/2Sr1/2MnO3 , as seen in Fig. 57(a). At low temperatures, there is a resistive change of six orders of magnitude in a field of 7 T. Kuwahara and co-workers (Kuwahara et al., 1995; Kuwahara and Tokura, 1998) have
mapped out the field-temperature phase diagram for this
compound by cycling the field at fixed temperature. As
seen in Fig. 57(b), the region of bistability grows dramatically as the temperature is reduced, with the metallic state tending toward metastability. Kuwahara and
Tokura (1998) suggest that there exist two local minima
in the free energy of the system, one corresponding to a
large-M, charge liquid state and the other a small-M,
charge ordered one, separated by a barrier U. At a critical value of the field, the Zeeman energy favors the
metastable, large-M state and the system becomes metallic. As the field is removed, the barrier prevents the
return of the system to its stable state, tending to trap it
in the metastable, conducting state.
When the hole concentration is not exactly 1/2 or 2/3
and, in the case of La1/2Ca1/2MnO3 , possibly even at x
⫽1/2 (Cheong and Chen, 1998), charge order is found to
be incommensurate. There is evidence for coexistence of
the x⫽1/2 and 2/3 structures as well as discommensurations. As a consequence of the discommensurations,
there is a spin slip associated with the Mn3⫹ sites that
results in a shorter correlation length for Mn3⫹ sublattice (Cheong and Chen, 1998). Figure 58 is a schematic
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 55. Dependence of charge-ordering transition on ionic
size or tolerance factor for various manganites. Phases include
paramagnetic insulator (PI), a charge-ordered insulator (COI),
and a ferromagnetic metal (FM), respectively. From Kuwahara
and Tokura, 1998.
sketch of a discommensuration that results from a spin
(and orbital) slip along a line of Mn4⫹ ions.
There have been several theoretical efforts to understand the charge ordering mechanism and the emergence of the CE antiferromagnetic state. A general
study of charge ordering in the context of the Holstein
model was carried out by Chiuchi and de Pasquale using
dynamical mean-field theory (Chiuchi and de Pasquale,
1999). They identify a phase boundary between a randomly distorted, charge ordered state and a polaronic
ordered phase as a function of the effective electronphonon coupling parameter ␭ (see Sec. IV). van
Veenendal and Fedro argue that when polarons encompass two sites, the hopping of the electron between them
FIG. 57. Field-induced metastability at x⫽( 12 ); (a) fieldinduced metal-insulator transition in Nd1/2Sr1/2MnO3 taken
from a series of Sm-doped samples From Tokura et al., 1996;
(b) field-temperature phase diagram for Nd1/2Sr1/2MnO3. The
hatched area corresponds to a region of metastability. From
Kuwahara et al., 1995.
‘‘dresses’’ the polaron, and provides an explanation for
the ferromagnet-CE phase transition (van Veenendaal
and Fedro, 1999). Shen and Wang (1999) demonstrate
that Wigner crystallization occurs within the double exchange model when processes that allow the possibility
of S-1/2 states, where S is the spin of the t 2g core, even
absent Jahn-Teller coupling. This may be relevant to the
Wigner crystal ordering observed at 2/3 filling.
B. Layered manganites
FIG. 56. Field-induced metal-insulator
Pr1/2Ca1/2MnO3. From Tomioka et al., 1996.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
As discussed in Sec. III.B.2, the bilayer manganites
RESr2Mn2O7 also show charge ordering effects. The
situation is more complicated with the charge-ordered
state exhibiting a competition between A-type and
(Nd1⫺x Lax )Sr2Mn2O7, Moritomo et al. (1999) found a
M. B. Salamon and M. Jaime: Manganites: Structure and transport
FIG. 58. Sketch of a discommensuration exhibiting a spin slip
in the Mn3⫹ sublattice. From Cheong and Chen, 1998.
narrow, charge-order CE antiferromagnetic regime to
exist on the La-rich end of the series, but to dominate in
a narrow temperature range between 110 and 220 K. On
cooling, the resistivity of LaSr2Mn2O7 first increases
sharply at the charge ordering transition, reaches a
maximum, and then decreases to a minimum near 100 K.
Single-crystal neutron-scattering data provide greater
detail (Kubota, Fujioka et al., 1999; Kubota, Yoshizawa
et al., 1999). As shown in Fig. 59, A-type antiferromagnetic order [Fig. 59(a)] and CE-type charge order [Fig.
59(b)] appear simultaneously at 210 K. This coincides
with a decrease in diffuse magnetic scattering [Fig.
59(c)]. Near 120 K, the CE-type charge ordering disappears and a weak, CE-type antiferromagnetic peak [Fig.
59(d)] appears. A small volume fraction of CE antiferromagnetic order persists to low temperature, but the
FIG. 59. Neutron-diffraction results for NdSr2Mn2O7. (a) Antiferromagnetism and (b) charge ordering appear at 210 K,
while (c) diffuse magnetic scattering decreases below that temperature. (d). Near 120 K, the antiferromagnetic order changes
to CE type over a narrow temperature range. From Kubota,
Fujioka et al., 1999b.
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
main contribution disappears at the temperature at
which CE charge order vanishes. The authors conclude
that this ‘‘melting’’ of the charge ordered state reflects
the competition between A-type d x 2 ⫺y 2 and CE-type
d 3x 2 ⫺r 2 and d 3y 2 ⫺r 2 orbitals. Substitution of Nd for La
lowers and reduces the resistivity peak, but leaves some
remanence of the charge ordered state in the temperature range between 100 and 200 K.
The resistivity of the single-layered compound
La1/2Sr3/2MnO4 increases slowly with decreasing temperature to approximately 240 K, below which it increases rapidly (Moritomo et al., 1995). Electron diffraction showed this temperature to mark the onset of
charge ordering. Subsequent work by Bao et al. (1996)
found the charge order superlattice spots to be consistent with 3D ordering, and to lie at 1/4 positions. However, neutron-scattering data (Sternlieb et al., 1996)
showed that charge order to set in near 220 K in the
pattern shown in Fig. 14, with spin ordering occurring at
110 K. Peaks at the 1/4 position were observed by x-ray
diffraction at the Mn K edge, and were attributed to
orbital ordering in the pattern shown in Fig. 14 (Murakami, Kawada et al., 1998).
Between the time of early experiments (Jonker, and
van Santen, 1950; van Santen and Jonker, 1950; Volger,
1954) and subsequent theoretical treatments (Zener,
1951; Anderson and Hasegawa, 1955; Goodenough,
1955; de Gennes, 1960) in the 1950s and the work of Jin
and co-workers in the 1990s (Jin, McCormack et al.,
1994; Jin, Tiefely et al., 1994) this field lay dormant. The
double exchange mechanism was partly at fault, lulling
researchers into believing that it contained the essential
elements needed to understand these materials. The reawakening of the field came as a consequence of the
large increases in magnetoresistance discovered in
ferromagnetic/nonmagnetic metallic multilayers, and
their ability to increase dramatically the areal bit density
in magnetic recording. However, we have seen the CMR
effect is only one manifestation of a competition between double exchange, that favors ferromagnetic order,
and a combination of Jahn-Teller coupling, Coulomb interactions, and antiferromagnetic superexchange, which
favor various flavors of charge/orbital/antiferromagnetic
order. The main impact of the double exchange mechanism is to serve as an amplifier of the magnetic field in
that alignment of neighboring spins serves as a valve
that controls the flow of doped-in holes through the
double exchange transfer matrix element t eff
⫽t cos(␪/2). The amplification can be readily seen in the
heat capacity of La0.7Ca0.3MnO3 near the ferromagnetic
transition, Fig. 44 where a field of 7 T, corresponding to
12 K for S⫽2, gives rise to a shift in the transition temperature more than three times larger. This even more
dramatically demonstrated in Fig. 57(a), where a 7-T
field drives the charge ordering transition from 150 K to
zero, a tenfold amplification of the field energy. Here
again, aligning spins delocalizes the doped-in hole, re-
M. B. Salamon and M. Jaime: Manganites: Structure and transport
duces its kinetic energy, and in turn reinforces the effect
of the field. In that way, a relatively small magnetic field
creates a band and with it, an energy gain of 3/5 of the
Fermi energy which, at half filling, is 3/10 of the bandwidth. Taking the bandwidth to be of order (104 K)k B
(Coey et al., 1999) this yields more than the 150 K required to balance the charge ordering energy of
The amplification of magnetic order described above
tends to overcome the tendency for charge/orbital ordering and to drive, thereby, the colossal magnetoresistance. How this comes about has been clarified by recent
neutron-scattering (Dai, Fernandez-Baca et al., 2000)
and electron-diffraction results (Zuo and Tao, 2001), on
La2/3Ca1/3MnO3 . Diffuse, nonmagnetic, quasielastic
peaks are observed in neutron scattering at (1/4, 1/4, 0)
positions, indicating the presence of short-range (1–2
nm) CE-type antiferromagnetic orbital ordering. Confirming data from electron diffraction reveals the presence of 1/2-order diffuse charge peaks of comparable
correlation length. The CE-type diffuse intensity increases with decreasing temperature, reaching a maximum near the temperature of the resistivity maximum.
At the same time, underlying diffuse scattering attributed to uncorrelated polarons, relatively constant above
T C , drops gradually below. To achieve such large volumes of fully charge ordered regions with alternating
Mn3⫹ and Mn4⫹ sites in a sample with only 1/3 of the
sites doped to Mn4⫹ would require excessive Coulomb
energy. Rather, there must be slightly hole-rich regions,
exhibiting CE-type antiferromagnetic charge modulations, surrounded by hole-poor regions. Because CEtype antiferromagnetic charge order is stripelike, the
manganites in this regime resemble, to some extent, the
‘‘electronic liquid crystal phases’’ that have been proposed for cuprates doped to comparable levels. As the
temperature is reduced, the hole-rich regions find themselves above the global Curie temperature, and develop
ferromagnetic correlations. These correlations then suppress the tendency toward charge/orbital order, cause
the diffuse CE-type peaks to lose intensity, and produce
local conducting regions, exactly as postulated by the
two-fluid model. This picture is also consistent with the
observation that ferromagnetic correlations do not grow
critically as the phase transition is approached, but
rather saturate at roughly the same 1–2-nm scale observed for the size of CE-correlated regions.
The competition between charge/orbital ordering and
ferromagnetism raises fundamental questions about the
nature of the transitions. Away from the ferromagnetic
concentration regime, the charge ordering transitions
are clearly first order in both field and temperature.
However, the ferromagnetic transitions in the ‘‘nominal’’
concentration regime are not obviously first order, as is
suggested by Fig. 45. However, as the heat-capacity data
show, they do not behave as expected for conventional
second-order transitions, either. As the bandwidth/band
filling point in Fig. 45 moves into the 360-K contour, the
nature of the phase transition becomes much closer to
that of a conventional ferromagnetic transition, as seen
Rev. Mod. Phys., Vol. 73, No. 3, July 2001
in the scaling curve of Fig. 46. This suggests that the
tendency to localize charge carriers into insulating and
ferromagnetic regimes causes the material to act like a
ferromagnet with quenched in disorder, and therefore
opens the possibility that the critical behavior is influenced by Griffiths singularities (Griffiths, 1969; Bray,
1987). Indeed, the spin dynamics in the region between
the observed ordering temperature (⬇250 K for
La2/3Ca1/3MnO3) and the highest temperature when all
double exchange bonds are active (⬇360 K for
La2/3Sr1/3MnO3) have been found to exhibit the nonexponential relaxation (Heffner et al., 1997, 2000) expected for Griffiths phases. As deduced from the resistivity, the magnetic transition occurs as regions with an
excess of active bonds percolate to develop long-range
Important theoretical challenges also remain in understanding the low-temperature properties. Despite differences in Curie temperature and doping, the underlying
exchange interactions in the ferromagnetic state appear
to be robust, at least as revealed by the behavior of longwavelength ferromagnetic spin waves seen in Fig. 14. It
remains to be understood how the exchange interaction
can be insensitive to the number of possible Mn3⫹-Mn4⫹
pairs. As pointed out in Sec. III.C.2, the resistivity in the
ferromagnetic regime loses its temperature dependence
at low temperatures, in violation of Mathiesson’s rule.
Weak temperature dependence is expected in a halfmetallic ferromagnet, due to the suppression of spin-flip
scattering. However, the very excitation of spin waves
eventually restores a down-spin band and with it, the
possibility of spin-flip scattering. Just how the down-spin
band develops and when spin-wave scattering returns
remains an open question, although a first attempt to
address this has been made (Furukawa, 2000). The nature of the orbital degrees of freedom in the ferromagnetic regime also remains an open question. It has generally be thought of (Tokura and Nagaosa, 2000) as an
‘‘orbital liquid,’’ but there have been some recent suggestions that a chiral orbital liquid might be possible
(Maezono and Nagaosa, 2000).
From a condensed-matter perspective the tendency
for stripe order to occur in both electron doped (e.g.,
Ca-rich) and hole-doped (e.g., La-rich) ends of the
phase diagram make these materials interesting testing
grounds for understanding the nature of liquid-crystallike phases in dilute electronic systems. The strong core
magnetism and its partner, strong Hund-rule coupling,
prevent the itinerant electrons from developing superconducting fluctuations such as is the case in cuprates.
However, these materials represent one end of a continuum of transition-metal oxides with perovskite-type
structures. All of them exhibit competition among
charge, magnetic, and orbital order. When these are
present, of equal strength, but not strong, as in the copper end point, superconductivity can occur. The manganites, along with cobaltates and nickelates, and probably
ruthenates, must be fully understood before the unusual
metallic behavior of the cuprates can be comprehended.
It may well be that the metallic state of all these oxides
M. B. Salamon and M. Jaime: Manganites: Structure and transport
represents a new, non-Fermi-liquid state of dense electron systems and that new theory of metals will be required.
As noted above, the initial motivation for reexamining this materials was to explore their possible applications to magnetic recording, sensing, and memory. The
half-metallic ground state suggested that very large magnetoresistance could be attained by fabricating tunnel
junctions. These efforts have been reviewed elsewhere.
(Ramesh et al., 1998). In a similar regard, there are numerous reviews whose focus has been on the role of
crystal chemistry and the microstructure of films grown
by various methods. Our focus here has been on the
variety of possible states and the transitions among
them. Whether or not these materials eventually find
useful applications, they have already expanded our understanding of what is a metal or an insulator, and what
characterizes the transitions between these states. The
lesson is that strong, but evenly balanced, interactions
lead to dramatic changes in physical properties and that
an understanding of how to treat such situations remains
to be found.
Work described in this paper was carried out in part
with the support of the Department of Energy through
the F. Seitz Materials Research Laboratory, Grant No.
DEFG02-91ER45439. Work at the National High Magnetic Field Laboratory was performed under the auspices of the National Science Foundation, the State of
Florida, and the U.S. Department of Energy.
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